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Research ArticleRepresenting and Counting the Subgroups of the Group Z
119898times Z
119899
Mario Hampejs1 Nicki Holighaus2 Laacuteszloacute Toacuteth34 and Christoph Wiesmeyr1
1 NuHAG Faculty of Mathematics University of Vienna Oskar Morgenstern Platz 1 1090 Vienna Austria2 Acoustics Research Institute Austrian Academy of Sciences Wohllebengasse 12-14 1040 Vienna Austria3 Department of Mathematics University of Pecs Ifjusag Utja 6 Pecs 7624 Hungary4 Institute of Mathematics University of Natural Resources and Life Sciences Gregor-Mendel-Straszlige 33 1180 Vienna Austria
Correspondence should be addressed to Laszlo Toth ltothgammattkptehu
Received 28 July 2014 Accepted 14 September 2014 Published 16 October 2014
Academic Editor Andrej Dujella
Copyright copy 2014 Mario Hampejs et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We deduce a simple representation and the invariant factor decompositions of the subgroups of the group Z119898times Z119899 where119898 and
119899 are arbitrary positive integers We obtain formulas for the total number of subgroups and the number of subgroups of a givenorder
1 Introduction
LetZ119898be the group of residue classesmodulo119898 and consider
the direct product 119866 = Z119898
times Z119899 where 119898 and 119899 are
arbitrary positive integersThis paper aims to deduce a simplerepresentation and the invariant factor decompositions ofthe subgroups of the group 119866 As consequences we deriveformulas for the number of certain types of subgroups of 119866including the total number 119904(119898 119899) of its subgroups and thenumber 119904
119896(119898 119899) of its subgroups of order 119896 (119896 | 119898119899)
Subgroups of Z times Z (sublattices of the two-dimensionalinteger lattice) and associated counting functions were con-sidered by several authors in pure and appliedmathematics Itis known for example that the number of subgroups of index119899 in Z times Z is 120590(119899) the sum of the (positive) divisors of 119899See for example [1 2] [3 item A001615] Although featuresof the subgroups of 119866 not only are interesting by their ownbut also have applications one of them is described below itseems that a synthesis on subgroups of 119866 cannot be found inthe literature
In the case 119898 = 119899 the subgroups of Z119899times Z119899play
an important role in numerical harmonic analysis morespecifically in the field of applied time-frequency analysisTime-frequency analysis attempts to investigate functionbehavior via a phase space representation given by theshort-time Fourier transform [4] The short-time Fouriercoefficients of a function 119891 are given by inner products
with translated modulations (or time-frequency shifts) ofa prototype function 119892 assumed to be well localized inphase space for example a Gaussian In applications thephase space corresponding to discrete finite functions (orvectors) belonging to C119899 is exactly Z
119899times Z119899 Concerned with
the question of reconstruction from samples of short-timeFourier transforms it has been found that when samplingon lattices that is subgroups of Z
119899times Z119899 the associated
analysis and reconstruction operators are particularly richin structure which in turn can be exploited for efficientimplementation (cf [5ndash7] and references therein) It is ofparticular interest to find subgroups in a certain range ofcardinality therefore a complete characterization of thesegroups helps choose the best one for the desired application
We recall that a finite Abelian group of order gt 1 has rank119903 if it is isomorphic toZ
1198991
timessdot sdot sdottimesZ119899119903
where 1198991 119899
119903isin N1
and 119899119895
| 119899119895+1
(1 le 119895 le 119903 minus 1) which is the invariantfactor decomposition of the given group Here the number 119903is uniquely determined and represents the minimal numberof generators of the group For general accounts on finiteAbelian groups see for example [8 9]
It is known that for every finite Abelian group theproblem of counting all subgroups and the subgroups of agiven order reduces to 119901-groups which follows from theproperties of the subgroup lattice of the group (see [10 11])In particular for 119866 = Z
119898times Z119899 this can be formulated as
follows Assume that gcd(119898 119899) gt 1 Then 119866 is an Abelian
Hindawi Publishing CorporationJournal of NumbersVolume 2014 Article ID 491428 6 pageshttpdxdoiorg1011552014491428
2 Journal of Numbers
group of rank two since 119866 ≃ Z119906times ZV where 119906 = gcd(119898 119899)
and V = lcm(119898 119899) Let 119906 = 1199011198861
1sdot sdot sdot 119901119886119903
119903and V = 119901
1198871
1sdot sdot sdot 119901119887119903
119903be
the prime power factorizations of 119906 and V respectively where0 le 119886119895le 119887119895(1 le 119895 le 119903) Then
119904 (119898 119899) =
119903
prod
119895=1
119904 (119901
119886119895
119895 119901
119887119895
119895) (1)
119904119896(119898 119899) =
119903
prod
119895=1
119904119896119895
(119901
119886119895
119895 119901
119887119895
119895) (2)
where 119896 = 1198961sdot sdot sdot 119896119903and 119896
119895= 119901
119888119895
119895with some exponents 0 le
119888119895le 119886119895+ 119887119895(1 le 119895 le 119903)
Now consider the 119901-group Z119901119886 times Z119901119887 where 0 le 119886 le 119887
This is of rank two for 1 le 119886 le 119887 One has the simple explicitformulae
119904 (119901119886
119901119887
)
=
(119887 minus 119886 + 1) 119901119886+2
minus (119887 minus 119886 minus 1) 119901119886+1
minus (119886 + 119887 + 3) 119901 + (119886 + 119887 + 1)
(119901 minus 1)2
(3)
119904119901119888 (119901119886
119901119887
) =
119901119888+1
minus 1
119901 minus 1
119888 le 119886 le 119887
119901119886+1
minus 1
119901 minus 1
119886 le 119888 le 119887
119901119886+119887minus119888+1
minus 1
119901 minus 1
119886 le 119887 le 119888 le 119886 + 119887
(4)
Formula (3) was derived by Calugareanu [12 Section 4]and recently by Petrillo [13 Proposition 2] using Goursatrsquoslemma for groups Tarnauceanu [14 Proposition 29] and [15Theorem 33] deduced (3) and (4) by a method based onproperties of certain attached matrices
Therefore 119904(119898 119899) and 119904119896(119898 119899) can be computed using (1)
(3) and (2) (4) respectively We deduce other formulas for119904(119898 119899) and 119904
119896(119898 119899) (Theorems 3 and 4) which generalize
(3) and (4) and put them in more compact forms These areconsequences of a simple representation of the subgroups of119866 = Z
119898timesZ119899 given inTheorem 1 This representation might
be known but the only source we could find is the paper[5] where only a special case is treated in a different formMore exactly in [5 Lemma 41] a representation for latticesin Z119899times Z119899of redundancy 2 that is subgroups of Z
119899times Z119899
having index 1198992 is given using matrices in Hermite normalform Theorem 2 gives the invariant factor decompositionsof the subgroups of 119866 We also consider the number ofcyclic subgroups of Z
119898times Z119899(Theorem 5) and the number
of subgroups of a given exponent in Z119899times Z119899(Theorem 8)
Our approach is elementary using only simple group-theoretic and number-theoretic arguments The proofs aregiven in Section 4
Throughout the paper we use the following notationsN = 1 2 N
0= 0 1 2 120591(119899) and 120590(119899) are the
number and the sum respectively of the positive divisors of119899 120595(119899) = 119899prod
119901|119899(1 + 1119901) is the Dedekind function 120596(119899)
stands for the number of distinct prime factors of 119899 120583 is theMobius function 120601 denotes Eulerrsquos totient function and 120577 isthe Riemann zeta function
s
b
a
1
1
0
0
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
Figure 1
2 Subgroups of Z119898times Z119899
The subgroups ofZ119898timesZ119899can be identified and visualized in
the plane with sublattices of the lattice Z119898times Z119899 Every two-
dimensional sublattice is generated by two basis vectors Forexample Figure 1 shows the subgroup ofZ
12timesZ12having the
basis vectors (3 0) and (1 2)This suggests the following representation of the sub-
groups
Theorem 1 For every119898 119899 isin N let
119868119898119899
= (119886 119887 119905) isin N2 times N0 119886 | 119898 119887 | 119899
0 le 119905 le gcd(119886 119899119887
) minus 1
(5)
and for (119886 119887 119905) isin 119868119898119899
define
119867119886119887119905
= (119894119886 +
119895119905119886
gcd (119886 119899119887)
119895119887)
0 le 119894 le
119898
119886
minus 1 0 le 119895 le
119899
119887
minus 1
(6)
Then 119867119886119887119905
is a subgroup of order 119898119899119886119887 of Z119898times Z119899and
the map (119886 119887 119905) 997891rarr 119867119886119887119905
is a bijection between the set 119868119898119899
andthe set of subgroups of Z
119898times Z119899
Note that for the subgroup 119867119886119887119905
the basis vectorsmentioned above are (119886 0) and (119904 119887) where
119904 =
119905119886
gcd (119886 119899119887) (7)
This notation for 119904 will be used also in the rest of thepaper Note also that in the case 119886 = 119898 119887 = 119899 the area of
Journal of Numbers 3
the parallelogram spanned by the basis vectors is 119886119887 exactlythe index of119867
119886119887119905
We say that a subgroup119867 = 119867119886119887119905
is a subproduct ofZ119898times
Z119899if 119867 = 119867
1times 1198672 where 119867
1and 119867
2are subgroups of Z
119898
and Z119899 respectively
Theorem 2 (i) The invariant factor decomposition of thesubgroup119867
119886119887119905is given by
119867119886119887119905
≃ Z120572times Z120573 (8)
where
120572 = (
119898
119886
119899
119887
119899119904
119886119887
) 120573 =
119898119899
a119887120572(9)
satisfying 120572 | 120573(ii) The exponent of the subgroup119867
119886119887119905is 120573
(iii) The subgroup119867119886119887119905
is cyclic if and only if 120572 = 1(iv) The subgroup119867
119886119887119905is a subproduct if and only if 119905 = 0
and 1198671198861198870
= Z119898119886
times Z119899119887
Here 1198671198861198870
is cyclic if and only ifgcd (119898119886 119899119887) = 1
For example for the subgroup represented by Figure 1 onehas 119898 = 119899 = 12 119886 = 3 119887 = 2 119904 = 1 120572 = 2 and 120573 = 12 andthis subgroup is isomorphic to Z
2times Z12 It is not cyclic and
is not a subproductAccording toTheorem 1 the number 119904(119898 119899) of subgroups
of Z119898times Z119899can be obtained by counting the elements of the
set 119868119898119899
We deduce the following
Theorem 3 For every119898 119899 isin N 119904(119898 119899) is given by
119904 (119898 119899) = sum
119886|119898 119887|119899
gcd (119886 119887) (10)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898
119889
) 120591 (
119899
119889
) (11)
= sum
119889|gcd(119898119899)119889120591 (
119898119899
1198892) (12)
Formula (10) is a special case of a formula representingthe number of all subgroups of a class of groups formed ascyclic extensions of cyclic groups deduced by Calhoun [16]and having a laborious proof Note that formula (10) is givenwithout proof in [3 item A054584]
Note also that the function (119898 119899) 997891rarr 119904(119898 119899) is represent-ing a multiplicative arithmetic function of two variables thatis 119904(119898119898
1015840
1198991198991015840
) = 119904(119898 119899)119904(1198981015840
1198991015840
) holds for any 119898 1198991198981015840
1198991015840
isin
N such that gcd(1198981198991198981015840
1198991015840
) = 1 This property which is inconcordance with (1) is a direct consequence of formula (10)See Section 5
Let119873(119886 119887 119888) denote the number of solutions (119909 119910 119911 119905) isinN4 of the system of equations 119909119910 = 119886 119911119905 = 119887 119909119911 = 119888
Theorem 4 For every 119896119898 119899 isin N such that 119896 | 119898119899
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119887119886=119896
gcd (119886 119887)(13)
= sum
119889|gcd(119896119898)119890| gcd (119896119899)119896|119889119890
120601(
119889119890
119896
)
(14)
= sum
119889|gcd(119898119899119896)120601 (119889)119873(
119898
119889
119899
119889
119896
119889
) (15)
The identities (3) and (4) can be easily deduced from eachof the identities given inTheorems 3 and 4 respectively
Theorem 5 Let119898 119899 isin N(i) The number 119888(119898 119899) of cyclic subgroups of Z
119898times Z119899is
given by
119888 (119898 119899) = sum
119886|119898119887|119899
gcd(119898119886119899119887)=1
gcd (119886 119887)(16)
= sum
119886|119898119887|119899
120601 (gcd (119886 119887)) (17)
= sum
119889|gcd(119898119899)(120583 lowast 120601) (119889) 120591 (
119898
119889
) 120591 (
119899
119889
) (18)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898119899
1198892) (19)
(ii)The number of subproducts ofZ119898timesZ119899is 120591(119898)120591(119899) and
the number of its cyclic subproducts is 120591(119898119899)
Formula (17) as a special case of an identity valid forarbitrary finite Abelian groups was derived by the thirdauthor [17 18] using different arguments The function(119898 119899) 997891rarr 119888(119898 119899) is also multiplicative
3 Subgroups of Z119899times Z119899
In the case119898 = 119899 which is of special interest in applicationsthe results given in the previous section can be easily usedWe point out that 119899 997891rarr 119904(119899) = 119904(119899 119899) and 119899 997891rarr 119888(119899) =
119888(119899 119899) are multiplicative arithmetic functions of a singlevariable (sequences [3 items A060724A060648]) They canbe written in the form of Dirichlet convolutions as shown bythe next corollaries
Corollary 6 For every 119899 isin N
119904 (119899) = sum
119889119890=119899
120601 (119889) 1205912
(119890)
= sum
119889119890=119899
119889 120591 (1198902
)
(20)
4 Journal of Numbers
Corollary 7 For every 119899 isin N
119888 (119899) = sum
119889119890=119899
1198892120596(119890)
(21)
= sum
119889119890=119899
120601 (119889) 120591 (1198902
) (22)
Further convolutional representations can also be givenfor example
119904 (119899) = sum
119889119890=119899
120591 (119889) 120595 (119890) 119888 (119899) = sum
119889|119899
120595 (119889) (23)
all of these follow from the Dirichlet-series representationsinfin
sum
119899=1
119904 (119899)
119899119911
=
1205773
(119911) 120577 (119911 minus 1)
120577 (2119911)
(24)
infin
sum
119899=1
119888 (119899)
119899119911
=
1205772
(119911) 120577 (119911 minus 1)
120577 (2119911)
(25)
valid for 119911 isin CR(119911) gt 2Observe that
119904 (119899) = sum
119889|119899
119888 (119889) (119899 isin N) (26)
which is a simple consequence of (23) or of (24) and (25) Italso follows from the next result
Theorem 8 For every 119899 120575 isin N with 120575 | 119899 the number ofsubgroups of exponent 120575 ofZ
119899timesZ119899equals the number of cyclic
subgroups of Z120575times Z120575
4 Proofs
Proof of Theorem 1 Let 119867 be a subgroup of 119866 = Z119898times Z119899
Consider the natural projection 1205872
119866 rarr Z119899given by
1205872(119909 119910) = 119910 Then 120587
2(119867) is a subgroup of Z
119899and there is
a unique divisor 119887 of 119899 such that 1205872(119867) = ⟨119887⟩ = 119895119887 0 le 119895 le
119899119887 minus 1 Let 119904 ge 0 be minimal such that (119904 119887) isin 119867Furthermore consider the natural inclusion 120580
1 Z119898
rarr
119866 given by 1205801(119909) = (119909 0)Then 120580minus1
1(119867) is a subgroup ofZ
119898and
there exists a unique divisor 119886 of119898 such that 120580minus11(119867) = ⟨119886⟩
We show that 119867 = (119894119886 + 119895119904 119895119887) 119894 119895 isin Z Indeed forevery 119894 119895 isin Z (119894119886+119895119904 119895119887) = 119894(119886 0)+119895(119904 119887) isin 119867 On the otherhand for every (119906 V) isin 119867 one has V isin 120587
2(119867) and hence there
is 119895 isin Z such that V = 119895119887 We obtain (119906 minus 119895119904 0) = (119906 V) minus119895(119904 119887) isin 119867 119906minus119895119904 isin 120580
minus1
1(119867) and there is 119894 isin Zwith 119906minus119895119904 = 119894119886
Here a necessary condition is that (119904119899119887 0) isin 119867 (obtainedfor 119894 = 0 and 119895 = 119899119887) that is 119886 | 119904119899b equivalent to119886gcd(119886 119899119887) | 119904 Clearly if this is verified then for the aboverepresentation of 119867 it is enough to take the values 0 le 119894 le
119898119886 minus 1 and 0 le 119895 le 119899119887 minus 1Also dividing 119904 by 119886 we have 119904 = 119886119902 + 119903 with 0 le 119903 lt 119886
and (119903 119887) = (119904 119887) minus 119902(119886 0) isin 119867 showing that 119904 lt 119886 byits minimality Hence 119904 = 119905119886gcd(119886 119899119887) with 0 le 119905 le
gcd(119886 119899119887) minus 1 Thus we obtain the given representationConversely every (119886 119887 119905) isin 119868
119898119899generates a subgroup
119867119886119887119905
of order 119898119899(119886119887) of Z119898
times Z119899and the proof is
complete
Proof of Theorem 2 (i)-(ii) We first determine the exponentof the subgroup 119867
119886119887119905 119867119886119887119905
is generated by (119886 0) and (119904 119887)hence its exponent is the least commonmultiple of the ordersof these two elements The order of (119886 0) is119898119886 To computethe order of (119904 119887) note that119898 | 119903119904 if and only if119898gcd(119898 119904) |
119903Thus the order of (119904 119887) is lcm(119898gcd(119898 119904) 119899119887)We deducethat the exponent of119867
119886119887119905is
lcm(
119898
119886
119898
gcd (119898 119904)
119899
119887
)
= lcm(
119898119899
119899119886
119898119899
119899gcd (119898 119904)
119898119899
119898119887
)
=
119898119899
gcd (119899119886 119899119898 119899119904 119898119887)
=
119898119899
gcd (119898119887 119899119886 119899119904)
= 120573
(27)
For every finite Abelian group the rank of a nontrivialsubgroup is at most the rank of the groupTherefore the rankof 119867119886119887t is 1 or 2 That is 119867
119886119887119905≃ Z119860times Z119861with certain
119860 119861 isin N such that 119860 | 119861 Here the exponent of 119867119886119887119905
equalsthat of Z
119860times Z119861 which is lcm(119860 119861) = 119861 Using (ii) already
proved we deduce that 119861 = 120573 Since the order of 119867119886119887119905
is119860119861 = 119898119899(119886119887) we have 119860 = 119898119899(119886119887120573) = 120572
(iii) According to (i) 119867119886119887119905
≃ Z120572timesZ120573 where 120572 | 120573 Hence
119867119886119887119905
is cyclic if and only if 120572 = 1(iv) The subgroups of Z
119898are of form 119894119886 0 le 119894 le 119898119886 minus
1 where 119886 | 119898 and the properties follow from (6) and (iii)
Proof of Theorem 3 By its definition the number of elementsof the set 119868
119898119899is
sum
119886|119898119887|119899
sum
0le119905legcd(119886119899119887)minus11
= sum
119886|119898119887|119899
gcd(119886 119899119887
) = sum
119886|119898119887|119899
gcd (119886 119887) (28)
representing 119904(119898 119899) This is formula (10)To obtain formula (11) apply the Gauss formula 119899 =
sum119889|119899
120601(119889) (119899 isin N) by writing the following
119904 (119898 119899) = sum
119886|119898119887|119899
sum
119889|gcd(119886119887)120601 (119889) = sum
119886119909=119898
119887119910=119899
sum
119889119894=119886
119889119895=119887
120601 (119889) = sum
119889119894119909=119898
119889119895119910=119899
120601 (119889)
= sum
119889119906=119898
119889V=119899
120601 (119889) sum
119894119909=119906
119895119910=V
1 = sum
119889119906=119898
119889V=119899
120601 (119889) 120591 (119906) 120591 (V)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898
119889
) 120591 (
119899
119889
)
(29)
Journal of Numbers 5
Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])
120591 (119898) 120591 (119899) = sum
119889|gcd(119898119899)120591 (
119898119899
1198892) (119898 119899 isin N) (30)
Proof of Theorem 4 According toTheorem 1
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119899119886119887=119896
gcd(119886 119899119887
) (31)
giving (13) which can be written by Gaussrsquo formula again as
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119887119886=119896
sum
119888|119886119888|119887
120601 (119888) = sum
119888119894119909=119898
119888119895119910=119899
119888119895119909=119896
120601 (119888)
= sum
119889119894=119898
119890119910=119899
sum
119888119909=119889
119888119895=119890
119888119895119909=119896
120601 (119888)
(32)
where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as
119904119896(119898 119899) = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888) sum
119894119909=119906
119895119910=V119895119909=119908
1 = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888)119873 (119906 V 119908)
= sum
119888|gcd(119898119899119896)120601 (119888)119873(
119898
119888
119899
119888
119896
119888
)
(33)
and the proof is complete
Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum
119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or
119899 gt 1
119888 (119898 119899) = sum
119886|119898119887|119899
sum
1le119904le119886
119886119887|119899119904
gcd(119898119886119899119887119899119904119886119887)=1
1 = sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119909119910119903)=1
1
= sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
sum
119890|gcd(119909119910119903)
120583 (119890) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) sum
1le119904le119886
119886gcd(119886119895)|119904
1
(34)
where the inner sum is gcd(119886 119895) Hence
119888 (119898 119899) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) gcd (119886 119895) (35)
Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905
we obtain
119888 (119898 119899) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum119890119894=119911
119887119890=119905
120583 (119890) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum
119890|gcd(119911119905)120583 (119890)
= sum
119886119911=119898
119895119905=119899
gcd(119911119905)=1
gcd (119886 119895)
(36)
which is (16)
The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3
(ii) For the subproducts 1198671198861198870
the values 119886 | 119898 and 119887 | 119899
can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis
sum
119886|119898
119887|119899
gcd(119898119886119899119887)=1
1 = sum
119886119909=119898
119887119910=119899
gcd(119909119910)=1
1 = sum
119886119909=119898
119887119910=119899
sum
119890|gcd(119909119910)
120583 (119890)
= sum
119890119860=119898
119890119861=119899
120583 (119890) 120591 (119860) 120591 (119861)
= sum
119890|gcd(119898119899)120583 (119890) 120591 (
119898
119890
) 120591 (
119899
119890
) = 120591 (119898119899)
(37)
by the inverse Busche-Ramanujan identity
Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z
119899times Z119899is
119864120575(119899) = sum
119886|119899119887|119899
sum
1le119904le119886
119886119887|119899119904
119899gcd(119886119887119904)=120575
1 = sum
119886119909=119899
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119886119887119904)=119899120575
1
(38)
Write 119886 = 1198861119899120575 119887 = 119887
1119899120575 and 119904 = 119904
1119899120575 with
gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of
Theorem 5(i) that
119864120575(119899) = sum
119890119894119909=120575
119890119895119910=120575
120583 (119890) gcd (119894 119910) (39)
which is exactly 119888(120575 120575) = 119888(120575) (cf (35))
5 Further Remarks
(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr
119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr
gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details
(2) Asymptotic formulas with sharp error terms for thesums sum
119898119899le119909119904(119898 119899) and sum
119898119899le119909119888(119898 119899) were given in
the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861
is cyclic if and only if 120580minus11(119862) and 120580
minus1
2(119862) have coprime
orders where 1205801and 1205802are the natural inclusions ([21
Theorem 42]) In the case 119860 = Z119898 119861 = Z
119899 and
119862 = 119867119886119887119905
one has 120580minus11(119862) = 119898119886 and 120580minus1
2(119862) =
gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867
119886119887119905given in Theorem 2(iii) can be
6 Journal of Numbers
obtained also in this way It turns out that regardingthe sublattice 119867
119886119887119905is cyclic if and only if the
numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic
(4) Note also the next formula for the number of cyclicsubgroups of Z
119899times Z119899 derived in [22 Example 2]
119888 (119899) = sum
lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)
where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain
sum
lcm(119889119890)=119899gcd (119889 119890)
= sum
ℓ119886119887=119899
gcd(119886119887)=1
ℓ = sum
ℓ119896=119899
ℓ sum
119886119887=119896
gcd(119886119887)=1
1 = sum
ℓ119896=119899
ℓ2120596(119896)
= 119888 (119899)
(41)
according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =
(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887
are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le
119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899
sum0le119904le119886minus1
1 = sum119886119887=119899
119886 = 120590(119899) mentioned inSection 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)
References
[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005
[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006
[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg
[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001
[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005
[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001
[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998
[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012
[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995
[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994
[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951
[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004
[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011
[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007
[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010
[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987
[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011
[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012
[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986
[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ
119898timesZ119899rdquo International Journal of NumberTheory
vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat
lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo
Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Numbers
group of rank two since 119866 ≃ Z119906times ZV where 119906 = gcd(119898 119899)
and V = lcm(119898 119899) Let 119906 = 1199011198861
1sdot sdot sdot 119901119886119903
119903and V = 119901
1198871
1sdot sdot sdot 119901119887119903
119903be
the prime power factorizations of 119906 and V respectively where0 le 119886119895le 119887119895(1 le 119895 le 119903) Then
119904 (119898 119899) =
119903
prod
119895=1
119904 (119901
119886119895
119895 119901
119887119895
119895) (1)
119904119896(119898 119899) =
119903
prod
119895=1
119904119896119895
(119901
119886119895
119895 119901
119887119895
119895) (2)
where 119896 = 1198961sdot sdot sdot 119896119903and 119896
119895= 119901
119888119895
119895with some exponents 0 le
119888119895le 119886119895+ 119887119895(1 le 119895 le 119903)
Now consider the 119901-group Z119901119886 times Z119901119887 where 0 le 119886 le 119887
This is of rank two for 1 le 119886 le 119887 One has the simple explicitformulae
119904 (119901119886
119901119887
)
=
(119887 minus 119886 + 1) 119901119886+2
minus (119887 minus 119886 minus 1) 119901119886+1
minus (119886 + 119887 + 3) 119901 + (119886 + 119887 + 1)
(119901 minus 1)2
(3)
119904119901119888 (119901119886
119901119887
) =
119901119888+1
minus 1
119901 minus 1
119888 le 119886 le 119887
119901119886+1
minus 1
119901 minus 1
119886 le 119888 le 119887
119901119886+119887minus119888+1
minus 1
119901 minus 1
119886 le 119887 le 119888 le 119886 + 119887
(4)
Formula (3) was derived by Calugareanu [12 Section 4]and recently by Petrillo [13 Proposition 2] using Goursatrsquoslemma for groups Tarnauceanu [14 Proposition 29] and [15Theorem 33] deduced (3) and (4) by a method based onproperties of certain attached matrices
Therefore 119904(119898 119899) and 119904119896(119898 119899) can be computed using (1)
(3) and (2) (4) respectively We deduce other formulas for119904(119898 119899) and 119904
119896(119898 119899) (Theorems 3 and 4) which generalize
(3) and (4) and put them in more compact forms These areconsequences of a simple representation of the subgroups of119866 = Z
119898timesZ119899 given inTheorem 1 This representation might
be known but the only source we could find is the paper[5] where only a special case is treated in a different formMore exactly in [5 Lemma 41] a representation for latticesin Z119899times Z119899of redundancy 2 that is subgroups of Z
119899times Z119899
having index 1198992 is given using matrices in Hermite normalform Theorem 2 gives the invariant factor decompositionsof the subgroups of 119866 We also consider the number ofcyclic subgroups of Z
119898times Z119899(Theorem 5) and the number
of subgroups of a given exponent in Z119899times Z119899(Theorem 8)
Our approach is elementary using only simple group-theoretic and number-theoretic arguments The proofs aregiven in Section 4
Throughout the paper we use the following notationsN = 1 2 N
0= 0 1 2 120591(119899) and 120590(119899) are the
number and the sum respectively of the positive divisors of119899 120595(119899) = 119899prod
119901|119899(1 + 1119901) is the Dedekind function 120596(119899)
stands for the number of distinct prime factors of 119899 120583 is theMobius function 120601 denotes Eulerrsquos totient function and 120577 isthe Riemann zeta function
s
b
a
1
1
0
0
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
Figure 1
2 Subgroups of Z119898times Z119899
The subgroups ofZ119898timesZ119899can be identified and visualized in
the plane with sublattices of the lattice Z119898times Z119899 Every two-
dimensional sublattice is generated by two basis vectors Forexample Figure 1 shows the subgroup ofZ
12timesZ12having the
basis vectors (3 0) and (1 2)This suggests the following representation of the sub-
groups
Theorem 1 For every119898 119899 isin N let
119868119898119899
= (119886 119887 119905) isin N2 times N0 119886 | 119898 119887 | 119899
0 le 119905 le gcd(119886 119899119887
) minus 1
(5)
and for (119886 119887 119905) isin 119868119898119899
define
119867119886119887119905
= (119894119886 +
119895119905119886
gcd (119886 119899119887)
119895119887)
0 le 119894 le
119898
119886
minus 1 0 le 119895 le
119899
119887
minus 1
(6)
Then 119867119886119887119905
is a subgroup of order 119898119899119886119887 of Z119898times Z119899and
the map (119886 119887 119905) 997891rarr 119867119886119887119905
is a bijection between the set 119868119898119899
andthe set of subgroups of Z
119898times Z119899
Note that for the subgroup 119867119886119887119905
the basis vectorsmentioned above are (119886 0) and (119904 119887) where
119904 =
119905119886
gcd (119886 119899119887) (7)
This notation for 119904 will be used also in the rest of thepaper Note also that in the case 119886 = 119898 119887 = 119899 the area of
Journal of Numbers 3
the parallelogram spanned by the basis vectors is 119886119887 exactlythe index of119867
119886119887119905
We say that a subgroup119867 = 119867119886119887119905
is a subproduct ofZ119898times
Z119899if 119867 = 119867
1times 1198672 where 119867
1and 119867
2are subgroups of Z
119898
and Z119899 respectively
Theorem 2 (i) The invariant factor decomposition of thesubgroup119867
119886119887119905is given by
119867119886119887119905
≃ Z120572times Z120573 (8)
where
120572 = (
119898
119886
119899
119887
119899119904
119886119887
) 120573 =
119898119899
a119887120572(9)
satisfying 120572 | 120573(ii) The exponent of the subgroup119867
119886119887119905is 120573
(iii) The subgroup119867119886119887119905
is cyclic if and only if 120572 = 1(iv) The subgroup119867
119886119887119905is a subproduct if and only if 119905 = 0
and 1198671198861198870
= Z119898119886
times Z119899119887
Here 1198671198861198870
is cyclic if and only ifgcd (119898119886 119899119887) = 1
For example for the subgroup represented by Figure 1 onehas 119898 = 119899 = 12 119886 = 3 119887 = 2 119904 = 1 120572 = 2 and 120573 = 12 andthis subgroup is isomorphic to Z
2times Z12 It is not cyclic and
is not a subproductAccording toTheorem 1 the number 119904(119898 119899) of subgroups
of Z119898times Z119899can be obtained by counting the elements of the
set 119868119898119899
We deduce the following
Theorem 3 For every119898 119899 isin N 119904(119898 119899) is given by
119904 (119898 119899) = sum
119886|119898 119887|119899
gcd (119886 119887) (10)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898
119889
) 120591 (
119899
119889
) (11)
= sum
119889|gcd(119898119899)119889120591 (
119898119899
1198892) (12)
Formula (10) is a special case of a formula representingthe number of all subgroups of a class of groups formed ascyclic extensions of cyclic groups deduced by Calhoun [16]and having a laborious proof Note that formula (10) is givenwithout proof in [3 item A054584]
Note also that the function (119898 119899) 997891rarr 119904(119898 119899) is represent-ing a multiplicative arithmetic function of two variables thatis 119904(119898119898
1015840
1198991198991015840
) = 119904(119898 119899)119904(1198981015840
1198991015840
) holds for any 119898 1198991198981015840
1198991015840
isin
N such that gcd(1198981198991198981015840
1198991015840
) = 1 This property which is inconcordance with (1) is a direct consequence of formula (10)See Section 5
Let119873(119886 119887 119888) denote the number of solutions (119909 119910 119911 119905) isinN4 of the system of equations 119909119910 = 119886 119911119905 = 119887 119909119911 = 119888
Theorem 4 For every 119896119898 119899 isin N such that 119896 | 119898119899
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119887119886=119896
gcd (119886 119887)(13)
= sum
119889|gcd(119896119898)119890| gcd (119896119899)119896|119889119890
120601(
119889119890
119896
)
(14)
= sum
119889|gcd(119898119899119896)120601 (119889)119873(
119898
119889
119899
119889
119896
119889
) (15)
The identities (3) and (4) can be easily deduced from eachof the identities given inTheorems 3 and 4 respectively
Theorem 5 Let119898 119899 isin N(i) The number 119888(119898 119899) of cyclic subgroups of Z
119898times Z119899is
given by
119888 (119898 119899) = sum
119886|119898119887|119899
gcd(119898119886119899119887)=1
gcd (119886 119887)(16)
= sum
119886|119898119887|119899
120601 (gcd (119886 119887)) (17)
= sum
119889|gcd(119898119899)(120583 lowast 120601) (119889) 120591 (
119898
119889
) 120591 (
119899
119889
) (18)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898119899
1198892) (19)
(ii)The number of subproducts ofZ119898timesZ119899is 120591(119898)120591(119899) and
the number of its cyclic subproducts is 120591(119898119899)
Formula (17) as a special case of an identity valid forarbitrary finite Abelian groups was derived by the thirdauthor [17 18] using different arguments The function(119898 119899) 997891rarr 119888(119898 119899) is also multiplicative
3 Subgroups of Z119899times Z119899
In the case119898 = 119899 which is of special interest in applicationsthe results given in the previous section can be easily usedWe point out that 119899 997891rarr 119904(119899) = 119904(119899 119899) and 119899 997891rarr 119888(119899) =
119888(119899 119899) are multiplicative arithmetic functions of a singlevariable (sequences [3 items A060724A060648]) They canbe written in the form of Dirichlet convolutions as shown bythe next corollaries
Corollary 6 For every 119899 isin N
119904 (119899) = sum
119889119890=119899
120601 (119889) 1205912
(119890)
= sum
119889119890=119899
119889 120591 (1198902
)
(20)
4 Journal of Numbers
Corollary 7 For every 119899 isin N
119888 (119899) = sum
119889119890=119899
1198892120596(119890)
(21)
= sum
119889119890=119899
120601 (119889) 120591 (1198902
) (22)
Further convolutional representations can also be givenfor example
119904 (119899) = sum
119889119890=119899
120591 (119889) 120595 (119890) 119888 (119899) = sum
119889|119899
120595 (119889) (23)
all of these follow from the Dirichlet-series representationsinfin
sum
119899=1
119904 (119899)
119899119911
=
1205773
(119911) 120577 (119911 minus 1)
120577 (2119911)
(24)
infin
sum
119899=1
119888 (119899)
119899119911
=
1205772
(119911) 120577 (119911 minus 1)
120577 (2119911)
(25)
valid for 119911 isin CR(119911) gt 2Observe that
119904 (119899) = sum
119889|119899
119888 (119889) (119899 isin N) (26)
which is a simple consequence of (23) or of (24) and (25) Italso follows from the next result
Theorem 8 For every 119899 120575 isin N with 120575 | 119899 the number ofsubgroups of exponent 120575 ofZ
119899timesZ119899equals the number of cyclic
subgroups of Z120575times Z120575
4 Proofs
Proof of Theorem 1 Let 119867 be a subgroup of 119866 = Z119898times Z119899
Consider the natural projection 1205872
119866 rarr Z119899given by
1205872(119909 119910) = 119910 Then 120587
2(119867) is a subgroup of Z
119899and there is
a unique divisor 119887 of 119899 such that 1205872(119867) = ⟨119887⟩ = 119895119887 0 le 119895 le
119899119887 minus 1 Let 119904 ge 0 be minimal such that (119904 119887) isin 119867Furthermore consider the natural inclusion 120580
1 Z119898
rarr
119866 given by 1205801(119909) = (119909 0)Then 120580minus1
1(119867) is a subgroup ofZ
119898and
there exists a unique divisor 119886 of119898 such that 120580minus11(119867) = ⟨119886⟩
We show that 119867 = (119894119886 + 119895119904 119895119887) 119894 119895 isin Z Indeed forevery 119894 119895 isin Z (119894119886+119895119904 119895119887) = 119894(119886 0)+119895(119904 119887) isin 119867 On the otherhand for every (119906 V) isin 119867 one has V isin 120587
2(119867) and hence there
is 119895 isin Z such that V = 119895119887 We obtain (119906 minus 119895119904 0) = (119906 V) minus119895(119904 119887) isin 119867 119906minus119895119904 isin 120580
minus1
1(119867) and there is 119894 isin Zwith 119906minus119895119904 = 119894119886
Here a necessary condition is that (119904119899119887 0) isin 119867 (obtainedfor 119894 = 0 and 119895 = 119899119887) that is 119886 | 119904119899b equivalent to119886gcd(119886 119899119887) | 119904 Clearly if this is verified then for the aboverepresentation of 119867 it is enough to take the values 0 le 119894 le
119898119886 minus 1 and 0 le 119895 le 119899119887 minus 1Also dividing 119904 by 119886 we have 119904 = 119886119902 + 119903 with 0 le 119903 lt 119886
and (119903 119887) = (119904 119887) minus 119902(119886 0) isin 119867 showing that 119904 lt 119886 byits minimality Hence 119904 = 119905119886gcd(119886 119899119887) with 0 le 119905 le
gcd(119886 119899119887) minus 1 Thus we obtain the given representationConversely every (119886 119887 119905) isin 119868
119898119899generates a subgroup
119867119886119887119905
of order 119898119899(119886119887) of Z119898
times Z119899and the proof is
complete
Proof of Theorem 2 (i)-(ii) We first determine the exponentof the subgroup 119867
119886119887119905 119867119886119887119905
is generated by (119886 0) and (119904 119887)hence its exponent is the least commonmultiple of the ordersof these two elements The order of (119886 0) is119898119886 To computethe order of (119904 119887) note that119898 | 119903119904 if and only if119898gcd(119898 119904) |
119903Thus the order of (119904 119887) is lcm(119898gcd(119898 119904) 119899119887)We deducethat the exponent of119867
119886119887119905is
lcm(
119898
119886
119898
gcd (119898 119904)
119899
119887
)
= lcm(
119898119899
119899119886
119898119899
119899gcd (119898 119904)
119898119899
119898119887
)
=
119898119899
gcd (119899119886 119899119898 119899119904 119898119887)
=
119898119899
gcd (119898119887 119899119886 119899119904)
= 120573
(27)
For every finite Abelian group the rank of a nontrivialsubgroup is at most the rank of the groupTherefore the rankof 119867119886119887t is 1 or 2 That is 119867
119886119887119905≃ Z119860times Z119861with certain
119860 119861 isin N such that 119860 | 119861 Here the exponent of 119867119886119887119905
equalsthat of Z
119860times Z119861 which is lcm(119860 119861) = 119861 Using (ii) already
proved we deduce that 119861 = 120573 Since the order of 119867119886119887119905
is119860119861 = 119898119899(119886119887) we have 119860 = 119898119899(119886119887120573) = 120572
(iii) According to (i) 119867119886119887119905
≃ Z120572timesZ120573 where 120572 | 120573 Hence
119867119886119887119905
is cyclic if and only if 120572 = 1(iv) The subgroups of Z
119898are of form 119894119886 0 le 119894 le 119898119886 minus
1 where 119886 | 119898 and the properties follow from (6) and (iii)
Proof of Theorem 3 By its definition the number of elementsof the set 119868
119898119899is
sum
119886|119898119887|119899
sum
0le119905legcd(119886119899119887)minus11
= sum
119886|119898119887|119899
gcd(119886 119899119887
) = sum
119886|119898119887|119899
gcd (119886 119887) (28)
representing 119904(119898 119899) This is formula (10)To obtain formula (11) apply the Gauss formula 119899 =
sum119889|119899
120601(119889) (119899 isin N) by writing the following
119904 (119898 119899) = sum
119886|119898119887|119899
sum
119889|gcd(119886119887)120601 (119889) = sum
119886119909=119898
119887119910=119899
sum
119889119894=119886
119889119895=119887
120601 (119889) = sum
119889119894119909=119898
119889119895119910=119899
120601 (119889)
= sum
119889119906=119898
119889V=119899
120601 (119889) sum
119894119909=119906
119895119910=V
1 = sum
119889119906=119898
119889V=119899
120601 (119889) 120591 (119906) 120591 (V)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898
119889
) 120591 (
119899
119889
)
(29)
Journal of Numbers 5
Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])
120591 (119898) 120591 (119899) = sum
119889|gcd(119898119899)120591 (
119898119899
1198892) (119898 119899 isin N) (30)
Proof of Theorem 4 According toTheorem 1
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119899119886119887=119896
gcd(119886 119899119887
) (31)
giving (13) which can be written by Gaussrsquo formula again as
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119887119886=119896
sum
119888|119886119888|119887
120601 (119888) = sum
119888119894119909=119898
119888119895119910=119899
119888119895119909=119896
120601 (119888)
= sum
119889119894=119898
119890119910=119899
sum
119888119909=119889
119888119895=119890
119888119895119909=119896
120601 (119888)
(32)
where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as
119904119896(119898 119899) = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888) sum
119894119909=119906
119895119910=V119895119909=119908
1 = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888)119873 (119906 V 119908)
= sum
119888|gcd(119898119899119896)120601 (119888)119873(
119898
119888
119899
119888
119896
119888
)
(33)
and the proof is complete
Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum
119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or
119899 gt 1
119888 (119898 119899) = sum
119886|119898119887|119899
sum
1le119904le119886
119886119887|119899119904
gcd(119898119886119899119887119899119904119886119887)=1
1 = sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119909119910119903)=1
1
= sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
sum
119890|gcd(119909119910119903)
120583 (119890) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) sum
1le119904le119886
119886gcd(119886119895)|119904
1
(34)
where the inner sum is gcd(119886 119895) Hence
119888 (119898 119899) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) gcd (119886 119895) (35)
Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905
we obtain
119888 (119898 119899) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum119890119894=119911
119887119890=119905
120583 (119890) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum
119890|gcd(119911119905)120583 (119890)
= sum
119886119911=119898
119895119905=119899
gcd(119911119905)=1
gcd (119886 119895)
(36)
which is (16)
The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3
(ii) For the subproducts 1198671198861198870
the values 119886 | 119898 and 119887 | 119899
can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis
sum
119886|119898
119887|119899
gcd(119898119886119899119887)=1
1 = sum
119886119909=119898
119887119910=119899
gcd(119909119910)=1
1 = sum
119886119909=119898
119887119910=119899
sum
119890|gcd(119909119910)
120583 (119890)
= sum
119890119860=119898
119890119861=119899
120583 (119890) 120591 (119860) 120591 (119861)
= sum
119890|gcd(119898119899)120583 (119890) 120591 (
119898
119890
) 120591 (
119899
119890
) = 120591 (119898119899)
(37)
by the inverse Busche-Ramanujan identity
Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z
119899times Z119899is
119864120575(119899) = sum
119886|119899119887|119899
sum
1le119904le119886
119886119887|119899119904
119899gcd(119886119887119904)=120575
1 = sum
119886119909=119899
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119886119887119904)=119899120575
1
(38)
Write 119886 = 1198861119899120575 119887 = 119887
1119899120575 and 119904 = 119904
1119899120575 with
gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of
Theorem 5(i) that
119864120575(119899) = sum
119890119894119909=120575
119890119895119910=120575
120583 (119890) gcd (119894 119910) (39)
which is exactly 119888(120575 120575) = 119888(120575) (cf (35))
5 Further Remarks
(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr
119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr
gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details
(2) Asymptotic formulas with sharp error terms for thesums sum
119898119899le119909119904(119898 119899) and sum
119898119899le119909119888(119898 119899) were given in
the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861
is cyclic if and only if 120580minus11(119862) and 120580
minus1
2(119862) have coprime
orders where 1205801and 1205802are the natural inclusions ([21
Theorem 42]) In the case 119860 = Z119898 119861 = Z
119899 and
119862 = 119867119886119887119905
one has 120580minus11(119862) = 119898119886 and 120580minus1
2(119862) =
gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867
119886119887119905given in Theorem 2(iii) can be
6 Journal of Numbers
obtained also in this way It turns out that regardingthe sublattice 119867
119886119887119905is cyclic if and only if the
numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic
(4) Note also the next formula for the number of cyclicsubgroups of Z
119899times Z119899 derived in [22 Example 2]
119888 (119899) = sum
lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)
where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain
sum
lcm(119889119890)=119899gcd (119889 119890)
= sum
ℓ119886119887=119899
gcd(119886119887)=1
ℓ = sum
ℓ119896=119899
ℓ sum
119886119887=119896
gcd(119886119887)=1
1 = sum
ℓ119896=119899
ℓ2120596(119896)
= 119888 (119899)
(41)
according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =
(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887
are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le
119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899
sum0le119904le119886minus1
1 = sum119886119887=119899
119886 = 120590(119899) mentioned inSection 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)
References
[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005
[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006
[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg
[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001
[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005
[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001
[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998
[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012
[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995
[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994
[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951
[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004
[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011
[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007
[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010
[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987
[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011
[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012
[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986
[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ
119898timesZ119899rdquo International Journal of NumberTheory
vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat
lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo
Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Numbers 3
the parallelogram spanned by the basis vectors is 119886119887 exactlythe index of119867
119886119887119905
We say that a subgroup119867 = 119867119886119887119905
is a subproduct ofZ119898times
Z119899if 119867 = 119867
1times 1198672 where 119867
1and 119867
2are subgroups of Z
119898
and Z119899 respectively
Theorem 2 (i) The invariant factor decomposition of thesubgroup119867
119886119887119905is given by
119867119886119887119905
≃ Z120572times Z120573 (8)
where
120572 = (
119898
119886
119899
119887
119899119904
119886119887
) 120573 =
119898119899
a119887120572(9)
satisfying 120572 | 120573(ii) The exponent of the subgroup119867
119886119887119905is 120573
(iii) The subgroup119867119886119887119905
is cyclic if and only if 120572 = 1(iv) The subgroup119867
119886119887119905is a subproduct if and only if 119905 = 0
and 1198671198861198870
= Z119898119886
times Z119899119887
Here 1198671198861198870
is cyclic if and only ifgcd (119898119886 119899119887) = 1
For example for the subgroup represented by Figure 1 onehas 119898 = 119899 = 12 119886 = 3 119887 = 2 119904 = 1 120572 = 2 and 120573 = 12 andthis subgroup is isomorphic to Z
2times Z12 It is not cyclic and
is not a subproductAccording toTheorem 1 the number 119904(119898 119899) of subgroups
of Z119898times Z119899can be obtained by counting the elements of the
set 119868119898119899
We deduce the following
Theorem 3 For every119898 119899 isin N 119904(119898 119899) is given by
119904 (119898 119899) = sum
119886|119898 119887|119899
gcd (119886 119887) (10)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898
119889
) 120591 (
119899
119889
) (11)
= sum
119889|gcd(119898119899)119889120591 (
119898119899
1198892) (12)
Formula (10) is a special case of a formula representingthe number of all subgroups of a class of groups formed ascyclic extensions of cyclic groups deduced by Calhoun [16]and having a laborious proof Note that formula (10) is givenwithout proof in [3 item A054584]
Note also that the function (119898 119899) 997891rarr 119904(119898 119899) is represent-ing a multiplicative arithmetic function of two variables thatis 119904(119898119898
1015840
1198991198991015840
) = 119904(119898 119899)119904(1198981015840
1198991015840
) holds for any 119898 1198991198981015840
1198991015840
isin
N such that gcd(1198981198991198981015840
1198991015840
) = 1 This property which is inconcordance with (1) is a direct consequence of formula (10)See Section 5
Let119873(119886 119887 119888) denote the number of solutions (119909 119910 119911 119905) isinN4 of the system of equations 119909119910 = 119886 119911119905 = 119887 119909119911 = 119888
Theorem 4 For every 119896119898 119899 isin N such that 119896 | 119898119899
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119887119886=119896
gcd (119886 119887)(13)
= sum
119889|gcd(119896119898)119890| gcd (119896119899)119896|119889119890
120601(
119889119890
119896
)
(14)
= sum
119889|gcd(119898119899119896)120601 (119889)119873(
119898
119889
119899
119889
119896
119889
) (15)
The identities (3) and (4) can be easily deduced from eachof the identities given inTheorems 3 and 4 respectively
Theorem 5 Let119898 119899 isin N(i) The number 119888(119898 119899) of cyclic subgroups of Z
119898times Z119899is
given by
119888 (119898 119899) = sum
119886|119898119887|119899
gcd(119898119886119899119887)=1
gcd (119886 119887)(16)
= sum
119886|119898119887|119899
120601 (gcd (119886 119887)) (17)
= sum
119889|gcd(119898119899)(120583 lowast 120601) (119889) 120591 (
119898
119889
) 120591 (
119899
119889
) (18)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898119899
1198892) (19)
(ii)The number of subproducts ofZ119898timesZ119899is 120591(119898)120591(119899) and
the number of its cyclic subproducts is 120591(119898119899)
Formula (17) as a special case of an identity valid forarbitrary finite Abelian groups was derived by the thirdauthor [17 18] using different arguments The function(119898 119899) 997891rarr 119888(119898 119899) is also multiplicative
3 Subgroups of Z119899times Z119899
In the case119898 = 119899 which is of special interest in applicationsthe results given in the previous section can be easily usedWe point out that 119899 997891rarr 119904(119899) = 119904(119899 119899) and 119899 997891rarr 119888(119899) =
119888(119899 119899) are multiplicative arithmetic functions of a singlevariable (sequences [3 items A060724A060648]) They canbe written in the form of Dirichlet convolutions as shown bythe next corollaries
Corollary 6 For every 119899 isin N
119904 (119899) = sum
119889119890=119899
120601 (119889) 1205912
(119890)
= sum
119889119890=119899
119889 120591 (1198902
)
(20)
4 Journal of Numbers
Corollary 7 For every 119899 isin N
119888 (119899) = sum
119889119890=119899
1198892120596(119890)
(21)
= sum
119889119890=119899
120601 (119889) 120591 (1198902
) (22)
Further convolutional representations can also be givenfor example
119904 (119899) = sum
119889119890=119899
120591 (119889) 120595 (119890) 119888 (119899) = sum
119889|119899
120595 (119889) (23)
all of these follow from the Dirichlet-series representationsinfin
sum
119899=1
119904 (119899)
119899119911
=
1205773
(119911) 120577 (119911 minus 1)
120577 (2119911)
(24)
infin
sum
119899=1
119888 (119899)
119899119911
=
1205772
(119911) 120577 (119911 minus 1)
120577 (2119911)
(25)
valid for 119911 isin CR(119911) gt 2Observe that
119904 (119899) = sum
119889|119899
119888 (119889) (119899 isin N) (26)
which is a simple consequence of (23) or of (24) and (25) Italso follows from the next result
Theorem 8 For every 119899 120575 isin N with 120575 | 119899 the number ofsubgroups of exponent 120575 ofZ
119899timesZ119899equals the number of cyclic
subgroups of Z120575times Z120575
4 Proofs
Proof of Theorem 1 Let 119867 be a subgroup of 119866 = Z119898times Z119899
Consider the natural projection 1205872
119866 rarr Z119899given by
1205872(119909 119910) = 119910 Then 120587
2(119867) is a subgroup of Z
119899and there is
a unique divisor 119887 of 119899 such that 1205872(119867) = ⟨119887⟩ = 119895119887 0 le 119895 le
119899119887 minus 1 Let 119904 ge 0 be minimal such that (119904 119887) isin 119867Furthermore consider the natural inclusion 120580
1 Z119898
rarr
119866 given by 1205801(119909) = (119909 0)Then 120580minus1
1(119867) is a subgroup ofZ
119898and
there exists a unique divisor 119886 of119898 such that 120580minus11(119867) = ⟨119886⟩
We show that 119867 = (119894119886 + 119895119904 119895119887) 119894 119895 isin Z Indeed forevery 119894 119895 isin Z (119894119886+119895119904 119895119887) = 119894(119886 0)+119895(119904 119887) isin 119867 On the otherhand for every (119906 V) isin 119867 one has V isin 120587
2(119867) and hence there
is 119895 isin Z such that V = 119895119887 We obtain (119906 minus 119895119904 0) = (119906 V) minus119895(119904 119887) isin 119867 119906minus119895119904 isin 120580
minus1
1(119867) and there is 119894 isin Zwith 119906minus119895119904 = 119894119886
Here a necessary condition is that (119904119899119887 0) isin 119867 (obtainedfor 119894 = 0 and 119895 = 119899119887) that is 119886 | 119904119899b equivalent to119886gcd(119886 119899119887) | 119904 Clearly if this is verified then for the aboverepresentation of 119867 it is enough to take the values 0 le 119894 le
119898119886 minus 1 and 0 le 119895 le 119899119887 minus 1Also dividing 119904 by 119886 we have 119904 = 119886119902 + 119903 with 0 le 119903 lt 119886
and (119903 119887) = (119904 119887) minus 119902(119886 0) isin 119867 showing that 119904 lt 119886 byits minimality Hence 119904 = 119905119886gcd(119886 119899119887) with 0 le 119905 le
gcd(119886 119899119887) minus 1 Thus we obtain the given representationConversely every (119886 119887 119905) isin 119868
119898119899generates a subgroup
119867119886119887119905
of order 119898119899(119886119887) of Z119898
times Z119899and the proof is
complete
Proof of Theorem 2 (i)-(ii) We first determine the exponentof the subgroup 119867
119886119887119905 119867119886119887119905
is generated by (119886 0) and (119904 119887)hence its exponent is the least commonmultiple of the ordersof these two elements The order of (119886 0) is119898119886 To computethe order of (119904 119887) note that119898 | 119903119904 if and only if119898gcd(119898 119904) |
119903Thus the order of (119904 119887) is lcm(119898gcd(119898 119904) 119899119887)We deducethat the exponent of119867
119886119887119905is
lcm(
119898
119886
119898
gcd (119898 119904)
119899
119887
)
= lcm(
119898119899
119899119886
119898119899
119899gcd (119898 119904)
119898119899
119898119887
)
=
119898119899
gcd (119899119886 119899119898 119899119904 119898119887)
=
119898119899
gcd (119898119887 119899119886 119899119904)
= 120573
(27)
For every finite Abelian group the rank of a nontrivialsubgroup is at most the rank of the groupTherefore the rankof 119867119886119887t is 1 or 2 That is 119867
119886119887119905≃ Z119860times Z119861with certain
119860 119861 isin N such that 119860 | 119861 Here the exponent of 119867119886119887119905
equalsthat of Z
119860times Z119861 which is lcm(119860 119861) = 119861 Using (ii) already
proved we deduce that 119861 = 120573 Since the order of 119867119886119887119905
is119860119861 = 119898119899(119886119887) we have 119860 = 119898119899(119886119887120573) = 120572
(iii) According to (i) 119867119886119887119905
≃ Z120572timesZ120573 where 120572 | 120573 Hence
119867119886119887119905
is cyclic if and only if 120572 = 1(iv) The subgroups of Z
119898are of form 119894119886 0 le 119894 le 119898119886 minus
1 where 119886 | 119898 and the properties follow from (6) and (iii)
Proof of Theorem 3 By its definition the number of elementsof the set 119868
119898119899is
sum
119886|119898119887|119899
sum
0le119905legcd(119886119899119887)minus11
= sum
119886|119898119887|119899
gcd(119886 119899119887
) = sum
119886|119898119887|119899
gcd (119886 119887) (28)
representing 119904(119898 119899) This is formula (10)To obtain formula (11) apply the Gauss formula 119899 =
sum119889|119899
120601(119889) (119899 isin N) by writing the following
119904 (119898 119899) = sum
119886|119898119887|119899
sum
119889|gcd(119886119887)120601 (119889) = sum
119886119909=119898
119887119910=119899
sum
119889119894=119886
119889119895=119887
120601 (119889) = sum
119889119894119909=119898
119889119895119910=119899
120601 (119889)
= sum
119889119906=119898
119889V=119899
120601 (119889) sum
119894119909=119906
119895119910=V
1 = sum
119889119906=119898
119889V=119899
120601 (119889) 120591 (119906) 120591 (V)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898
119889
) 120591 (
119899
119889
)
(29)
Journal of Numbers 5
Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])
120591 (119898) 120591 (119899) = sum
119889|gcd(119898119899)120591 (
119898119899
1198892) (119898 119899 isin N) (30)
Proof of Theorem 4 According toTheorem 1
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119899119886119887=119896
gcd(119886 119899119887
) (31)
giving (13) which can be written by Gaussrsquo formula again as
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119887119886=119896
sum
119888|119886119888|119887
120601 (119888) = sum
119888119894119909=119898
119888119895119910=119899
119888119895119909=119896
120601 (119888)
= sum
119889119894=119898
119890119910=119899
sum
119888119909=119889
119888119895=119890
119888119895119909=119896
120601 (119888)
(32)
where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as
119904119896(119898 119899) = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888) sum
119894119909=119906
119895119910=V119895119909=119908
1 = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888)119873 (119906 V 119908)
= sum
119888|gcd(119898119899119896)120601 (119888)119873(
119898
119888
119899
119888
119896
119888
)
(33)
and the proof is complete
Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum
119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or
119899 gt 1
119888 (119898 119899) = sum
119886|119898119887|119899
sum
1le119904le119886
119886119887|119899119904
gcd(119898119886119899119887119899119904119886119887)=1
1 = sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119909119910119903)=1
1
= sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
sum
119890|gcd(119909119910119903)
120583 (119890) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) sum
1le119904le119886
119886gcd(119886119895)|119904
1
(34)
where the inner sum is gcd(119886 119895) Hence
119888 (119898 119899) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) gcd (119886 119895) (35)
Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905
we obtain
119888 (119898 119899) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum119890119894=119911
119887119890=119905
120583 (119890) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum
119890|gcd(119911119905)120583 (119890)
= sum
119886119911=119898
119895119905=119899
gcd(119911119905)=1
gcd (119886 119895)
(36)
which is (16)
The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3
(ii) For the subproducts 1198671198861198870
the values 119886 | 119898 and 119887 | 119899
can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis
sum
119886|119898
119887|119899
gcd(119898119886119899119887)=1
1 = sum
119886119909=119898
119887119910=119899
gcd(119909119910)=1
1 = sum
119886119909=119898
119887119910=119899
sum
119890|gcd(119909119910)
120583 (119890)
= sum
119890119860=119898
119890119861=119899
120583 (119890) 120591 (119860) 120591 (119861)
= sum
119890|gcd(119898119899)120583 (119890) 120591 (
119898
119890
) 120591 (
119899
119890
) = 120591 (119898119899)
(37)
by the inverse Busche-Ramanujan identity
Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z
119899times Z119899is
119864120575(119899) = sum
119886|119899119887|119899
sum
1le119904le119886
119886119887|119899119904
119899gcd(119886119887119904)=120575
1 = sum
119886119909=119899
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119886119887119904)=119899120575
1
(38)
Write 119886 = 1198861119899120575 119887 = 119887
1119899120575 and 119904 = 119904
1119899120575 with
gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of
Theorem 5(i) that
119864120575(119899) = sum
119890119894119909=120575
119890119895119910=120575
120583 (119890) gcd (119894 119910) (39)
which is exactly 119888(120575 120575) = 119888(120575) (cf (35))
5 Further Remarks
(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr
119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr
gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details
(2) Asymptotic formulas with sharp error terms for thesums sum
119898119899le119909119904(119898 119899) and sum
119898119899le119909119888(119898 119899) were given in
the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861
is cyclic if and only if 120580minus11(119862) and 120580
minus1
2(119862) have coprime
orders where 1205801and 1205802are the natural inclusions ([21
Theorem 42]) In the case 119860 = Z119898 119861 = Z
119899 and
119862 = 119867119886119887119905
one has 120580minus11(119862) = 119898119886 and 120580minus1
2(119862) =
gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867
119886119887119905given in Theorem 2(iii) can be
6 Journal of Numbers
obtained also in this way It turns out that regardingthe sublattice 119867
119886119887119905is cyclic if and only if the
numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic
(4) Note also the next formula for the number of cyclicsubgroups of Z
119899times Z119899 derived in [22 Example 2]
119888 (119899) = sum
lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)
where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain
sum
lcm(119889119890)=119899gcd (119889 119890)
= sum
ℓ119886119887=119899
gcd(119886119887)=1
ℓ = sum
ℓ119896=119899
ℓ sum
119886119887=119896
gcd(119886119887)=1
1 = sum
ℓ119896=119899
ℓ2120596(119896)
= 119888 (119899)
(41)
according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =
(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887
are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le
119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899
sum0le119904le119886minus1
1 = sum119886119887=119899
119886 = 120590(119899) mentioned inSection 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)
References
[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005
[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006
[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg
[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001
[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005
[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001
[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998
[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012
[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995
[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994
[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951
[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004
[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011
[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007
[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010
[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987
[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011
[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012
[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986
[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ
119898timesZ119899rdquo International Journal of NumberTheory
vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat
lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo
Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Numbers
Corollary 7 For every 119899 isin N
119888 (119899) = sum
119889119890=119899
1198892120596(119890)
(21)
= sum
119889119890=119899
120601 (119889) 120591 (1198902
) (22)
Further convolutional representations can also be givenfor example
119904 (119899) = sum
119889119890=119899
120591 (119889) 120595 (119890) 119888 (119899) = sum
119889|119899
120595 (119889) (23)
all of these follow from the Dirichlet-series representationsinfin
sum
119899=1
119904 (119899)
119899119911
=
1205773
(119911) 120577 (119911 minus 1)
120577 (2119911)
(24)
infin
sum
119899=1
119888 (119899)
119899119911
=
1205772
(119911) 120577 (119911 minus 1)
120577 (2119911)
(25)
valid for 119911 isin CR(119911) gt 2Observe that
119904 (119899) = sum
119889|119899
119888 (119889) (119899 isin N) (26)
which is a simple consequence of (23) or of (24) and (25) Italso follows from the next result
Theorem 8 For every 119899 120575 isin N with 120575 | 119899 the number ofsubgroups of exponent 120575 ofZ
119899timesZ119899equals the number of cyclic
subgroups of Z120575times Z120575
4 Proofs
Proof of Theorem 1 Let 119867 be a subgroup of 119866 = Z119898times Z119899
Consider the natural projection 1205872
119866 rarr Z119899given by
1205872(119909 119910) = 119910 Then 120587
2(119867) is a subgroup of Z
119899and there is
a unique divisor 119887 of 119899 such that 1205872(119867) = ⟨119887⟩ = 119895119887 0 le 119895 le
119899119887 minus 1 Let 119904 ge 0 be minimal such that (119904 119887) isin 119867Furthermore consider the natural inclusion 120580
1 Z119898
rarr
119866 given by 1205801(119909) = (119909 0)Then 120580minus1
1(119867) is a subgroup ofZ
119898and
there exists a unique divisor 119886 of119898 such that 120580minus11(119867) = ⟨119886⟩
We show that 119867 = (119894119886 + 119895119904 119895119887) 119894 119895 isin Z Indeed forevery 119894 119895 isin Z (119894119886+119895119904 119895119887) = 119894(119886 0)+119895(119904 119887) isin 119867 On the otherhand for every (119906 V) isin 119867 one has V isin 120587
2(119867) and hence there
is 119895 isin Z such that V = 119895119887 We obtain (119906 minus 119895119904 0) = (119906 V) minus119895(119904 119887) isin 119867 119906minus119895119904 isin 120580
minus1
1(119867) and there is 119894 isin Zwith 119906minus119895119904 = 119894119886
Here a necessary condition is that (119904119899119887 0) isin 119867 (obtainedfor 119894 = 0 and 119895 = 119899119887) that is 119886 | 119904119899b equivalent to119886gcd(119886 119899119887) | 119904 Clearly if this is verified then for the aboverepresentation of 119867 it is enough to take the values 0 le 119894 le
119898119886 minus 1 and 0 le 119895 le 119899119887 minus 1Also dividing 119904 by 119886 we have 119904 = 119886119902 + 119903 with 0 le 119903 lt 119886
and (119903 119887) = (119904 119887) minus 119902(119886 0) isin 119867 showing that 119904 lt 119886 byits minimality Hence 119904 = 119905119886gcd(119886 119899119887) with 0 le 119905 le
gcd(119886 119899119887) minus 1 Thus we obtain the given representationConversely every (119886 119887 119905) isin 119868
119898119899generates a subgroup
119867119886119887119905
of order 119898119899(119886119887) of Z119898
times Z119899and the proof is
complete
Proof of Theorem 2 (i)-(ii) We first determine the exponentof the subgroup 119867
119886119887119905 119867119886119887119905
is generated by (119886 0) and (119904 119887)hence its exponent is the least commonmultiple of the ordersof these two elements The order of (119886 0) is119898119886 To computethe order of (119904 119887) note that119898 | 119903119904 if and only if119898gcd(119898 119904) |
119903Thus the order of (119904 119887) is lcm(119898gcd(119898 119904) 119899119887)We deducethat the exponent of119867
119886119887119905is
lcm(
119898
119886
119898
gcd (119898 119904)
119899
119887
)
= lcm(
119898119899
119899119886
119898119899
119899gcd (119898 119904)
119898119899
119898119887
)
=
119898119899
gcd (119899119886 119899119898 119899119904 119898119887)
=
119898119899
gcd (119898119887 119899119886 119899119904)
= 120573
(27)
For every finite Abelian group the rank of a nontrivialsubgroup is at most the rank of the groupTherefore the rankof 119867119886119887t is 1 or 2 That is 119867
119886119887119905≃ Z119860times Z119861with certain
119860 119861 isin N such that 119860 | 119861 Here the exponent of 119867119886119887119905
equalsthat of Z
119860times Z119861 which is lcm(119860 119861) = 119861 Using (ii) already
proved we deduce that 119861 = 120573 Since the order of 119867119886119887119905
is119860119861 = 119898119899(119886119887) we have 119860 = 119898119899(119886119887120573) = 120572
(iii) According to (i) 119867119886119887119905
≃ Z120572timesZ120573 where 120572 | 120573 Hence
119867119886119887119905
is cyclic if and only if 120572 = 1(iv) The subgroups of Z
119898are of form 119894119886 0 le 119894 le 119898119886 minus
1 where 119886 | 119898 and the properties follow from (6) and (iii)
Proof of Theorem 3 By its definition the number of elementsof the set 119868
119898119899is
sum
119886|119898119887|119899
sum
0le119905legcd(119886119899119887)minus11
= sum
119886|119898119887|119899
gcd(119886 119899119887
) = sum
119886|119898119887|119899
gcd (119886 119887) (28)
representing 119904(119898 119899) This is formula (10)To obtain formula (11) apply the Gauss formula 119899 =
sum119889|119899
120601(119889) (119899 isin N) by writing the following
119904 (119898 119899) = sum
119886|119898119887|119899
sum
119889|gcd(119886119887)120601 (119889) = sum
119886119909=119898
119887119910=119899
sum
119889119894=119886
119889119895=119887
120601 (119889) = sum
119889119894119909=119898
119889119895119910=119899
120601 (119889)
= sum
119889119906=119898
119889V=119899
120601 (119889) sum
119894119909=119906
119895119910=V
1 = sum
119889119906=119898
119889V=119899
120601 (119889) 120591 (119906) 120591 (V)
= sum
119889|gcd(119898119899)120601 (119889) 120591 (
119898
119889
) 120591 (
119899
119889
)
(29)
Journal of Numbers 5
Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])
120591 (119898) 120591 (119899) = sum
119889|gcd(119898119899)120591 (
119898119899
1198892) (119898 119899 isin N) (30)
Proof of Theorem 4 According toTheorem 1
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119899119886119887=119896
gcd(119886 119899119887
) (31)
giving (13) which can be written by Gaussrsquo formula again as
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119887119886=119896
sum
119888|119886119888|119887
120601 (119888) = sum
119888119894119909=119898
119888119895119910=119899
119888119895119909=119896
120601 (119888)
= sum
119889119894=119898
119890119910=119899
sum
119888119909=119889
119888119895=119890
119888119895119909=119896
120601 (119888)
(32)
where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as
119904119896(119898 119899) = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888) sum
119894119909=119906
119895119910=V119895119909=119908
1 = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888)119873 (119906 V 119908)
= sum
119888|gcd(119898119899119896)120601 (119888)119873(
119898
119888
119899
119888
119896
119888
)
(33)
and the proof is complete
Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum
119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or
119899 gt 1
119888 (119898 119899) = sum
119886|119898119887|119899
sum
1le119904le119886
119886119887|119899119904
gcd(119898119886119899119887119899119904119886119887)=1
1 = sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119909119910119903)=1
1
= sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
sum
119890|gcd(119909119910119903)
120583 (119890) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) sum
1le119904le119886
119886gcd(119886119895)|119904
1
(34)
where the inner sum is gcd(119886 119895) Hence
119888 (119898 119899) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) gcd (119886 119895) (35)
Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905
we obtain
119888 (119898 119899) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum119890119894=119911
119887119890=119905
120583 (119890) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum
119890|gcd(119911119905)120583 (119890)
= sum
119886119911=119898
119895119905=119899
gcd(119911119905)=1
gcd (119886 119895)
(36)
which is (16)
The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3
(ii) For the subproducts 1198671198861198870
the values 119886 | 119898 and 119887 | 119899
can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis
sum
119886|119898
119887|119899
gcd(119898119886119899119887)=1
1 = sum
119886119909=119898
119887119910=119899
gcd(119909119910)=1
1 = sum
119886119909=119898
119887119910=119899
sum
119890|gcd(119909119910)
120583 (119890)
= sum
119890119860=119898
119890119861=119899
120583 (119890) 120591 (119860) 120591 (119861)
= sum
119890|gcd(119898119899)120583 (119890) 120591 (
119898
119890
) 120591 (
119899
119890
) = 120591 (119898119899)
(37)
by the inverse Busche-Ramanujan identity
Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z
119899times Z119899is
119864120575(119899) = sum
119886|119899119887|119899
sum
1le119904le119886
119886119887|119899119904
119899gcd(119886119887119904)=120575
1 = sum
119886119909=119899
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119886119887119904)=119899120575
1
(38)
Write 119886 = 1198861119899120575 119887 = 119887
1119899120575 and 119904 = 119904
1119899120575 with
gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of
Theorem 5(i) that
119864120575(119899) = sum
119890119894119909=120575
119890119895119910=120575
120583 (119890) gcd (119894 119910) (39)
which is exactly 119888(120575 120575) = 119888(120575) (cf (35))
5 Further Remarks
(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr
119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr
gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details
(2) Asymptotic formulas with sharp error terms for thesums sum
119898119899le119909119904(119898 119899) and sum
119898119899le119909119888(119898 119899) were given in
the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861
is cyclic if and only if 120580minus11(119862) and 120580
minus1
2(119862) have coprime
orders where 1205801and 1205802are the natural inclusions ([21
Theorem 42]) In the case 119860 = Z119898 119861 = Z
119899 and
119862 = 119867119886119887119905
one has 120580minus11(119862) = 119898119886 and 120580minus1
2(119862) =
gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867
119886119887119905given in Theorem 2(iii) can be
6 Journal of Numbers
obtained also in this way It turns out that regardingthe sublattice 119867
119886119887119905is cyclic if and only if the
numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic
(4) Note also the next formula for the number of cyclicsubgroups of Z
119899times Z119899 derived in [22 Example 2]
119888 (119899) = sum
lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)
where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain
sum
lcm(119889119890)=119899gcd (119889 119890)
= sum
ℓ119886119887=119899
gcd(119886119887)=1
ℓ = sum
ℓ119896=119899
ℓ sum
119886119887=119896
gcd(119886119887)=1
1 = sum
ℓ119896=119899
ℓ2120596(119896)
= 119888 (119899)
(41)
according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =
(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887
are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le
119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899
sum0le119904le119886minus1
1 = sum119886119887=119899
119886 = 120590(119899) mentioned inSection 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)
References
[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005
[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006
[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg
[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001
[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005
[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001
[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998
[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012
[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995
[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994
[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951
[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004
[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011
[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007
[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010
[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987
[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011
[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012
[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986
[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ
119898timesZ119899rdquo International Journal of NumberTheory
vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat
lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo
Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Numbers 5
Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])
120591 (119898) 120591 (119899) = sum
119889|gcd(119898119899)120591 (
119898119899
1198892) (119898 119899 isin N) (30)
Proof of Theorem 4 According toTheorem 1
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119899119886119887=119896
gcd(119886 119899119887
) (31)
giving (13) which can be written by Gaussrsquo formula again as
119904119896(119898 119899) = sum
119886|119898119887|119899
119898119887119886=119896
sum
119888|119886119888|119887
120601 (119888) = sum
119888119894119909=119898
119888119895119910=119899
119888119895119909=119896
120601 (119888)
= sum
119889119894=119898
119890119910=119899
sum
119888119909=119889
119888119895=119890
119888119895119909=119896
120601 (119888)
(32)
where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as
119904119896(119898 119899) = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888) sum
119894119909=119906
119895119910=V119895119909=119908
1 = sum
119888119906=119898
119888V=119899119888119908=119896
120601 (119888)119873 (119906 V 119908)
= sum
119888|gcd(119898119899119896)120601 (119888)119873(
119898
119888
119899
119888
119896
119888
)
(33)
and the proof is complete
Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum
119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or
119899 gt 1
119888 (119898 119899) = sum
119886|119898119887|119899
sum
1le119904le119886
119886119887|119899119904
gcd(119898119886119899119887119899119904119886119887)=1
1 = sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119909119910119903)=1
1
= sum
119886119909=119898
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
sum
119890|gcd(119909119910119903)
120583 (119890) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) sum
1le119904le119886
119886gcd(119886119895)|119904
1
(34)
where the inner sum is gcd(119886 119895) Hence
119888 (119898 119899) = sum
119886119890119894=119898
119887119890119895=119899
120583 (119890) gcd (119886 119895) (35)
Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905
we obtain
119888 (119898 119899) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum119890119894=119911
119887119890=119905
120583 (119890) = sum
119886119911=119898
119895119905=119899
gcd (119886 119895) sum
119890|gcd(119911119905)120583 (119890)
= sum
119886119911=119898
119895119905=119899
gcd(119911119905)=1
gcd (119886 119895)
(36)
which is (16)
The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3
(ii) For the subproducts 1198671198861198870
the values 119886 | 119898 and 119887 | 119899
can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis
sum
119886|119898
119887|119899
gcd(119898119886119899119887)=1
1 = sum
119886119909=119898
119887119910=119899
gcd(119909119910)=1
1 = sum
119886119909=119898
119887119910=119899
sum
119890|gcd(119909119910)
120583 (119890)
= sum
119890119860=119898
119890119861=119899
120583 (119890) 120591 (119860) 120591 (119861)
= sum
119890|gcd(119898119899)120583 (119890) 120591 (
119898
119890
) 120591 (
119899
119890
) = 120591 (119898119899)
(37)
by the inverse Busche-Ramanujan identity
Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z
119899times Z119899is
119864120575(119899) = sum
119886|119899119887|119899
sum
1le119904le119886
119886119887|119899119904
119899gcd(119886119887119904)=120575
1 = sum
119886119909=119899
119887119910=119899
sum
1le119904le119886
119886119903=119910119904
gcd(119886119887119904)=119899120575
1
(38)
Write 119886 = 1198861119899120575 119887 = 119887
1119899120575 and 119904 = 119904
1119899120575 with
gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of
Theorem 5(i) that
119864120575(119899) = sum
119890119894119909=120575
119890119895119910=120575
120583 (119890) gcd (119894 119910) (39)
which is exactly 119888(120575 120575) = 119888(120575) (cf (35))
5 Further Remarks
(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr
119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr
gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details
(2) Asymptotic formulas with sharp error terms for thesums sum
119898119899le119909119904(119898 119899) and sum
119898119899le119909119888(119898 119899) were given in
the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861
is cyclic if and only if 120580minus11(119862) and 120580
minus1
2(119862) have coprime
orders where 1205801and 1205802are the natural inclusions ([21
Theorem 42]) In the case 119860 = Z119898 119861 = Z
119899 and
119862 = 119867119886119887119905
one has 120580minus11(119862) = 119898119886 and 120580minus1
2(119862) =
gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867
119886119887119905given in Theorem 2(iii) can be
6 Journal of Numbers
obtained also in this way It turns out that regardingthe sublattice 119867
119886119887119905is cyclic if and only if the
numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic
(4) Note also the next formula for the number of cyclicsubgroups of Z
119899times Z119899 derived in [22 Example 2]
119888 (119899) = sum
lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)
where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain
sum
lcm(119889119890)=119899gcd (119889 119890)
= sum
ℓ119886119887=119899
gcd(119886119887)=1
ℓ = sum
ℓ119896=119899
ℓ sum
119886119887=119896
gcd(119886119887)=1
1 = sum
ℓ119896=119899
ℓ2120596(119896)
= 119888 (119899)
(41)
according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =
(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887
are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le
119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899
sum0le119904le119886minus1
1 = sum119886119887=119899
119886 = 120590(119899) mentioned inSection 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)
References
[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005
[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006
[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg
[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001
[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005
[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001
[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998
[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012
[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995
[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994
[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951
[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004
[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011
[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007
[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010
[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987
[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011
[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012
[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986
[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ
119898timesZ119899rdquo International Journal of NumberTheory
vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat
lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo
Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Numbers
obtained also in this way It turns out that regardingthe sublattice 119867
119886119887119905is cyclic if and only if the
numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic
(4) Note also the next formula for the number of cyclicsubgroups of Z
119899times Z119899 derived in [22 Example 2]
119888 (119899) = sum
lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)
where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain
sum
lcm(119889119890)=119899gcd (119889 119890)
= sum
ℓ119886119887=119899
gcd(119886119887)=1
ℓ = sum
ℓ119896=119899
ℓ sum
119886119887=119896
gcd(119886119887)=1
1 = sum
ℓ119896=119899
ℓ2120596(119896)
= 119888 (119899)
(41)
according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =
(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887
are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le
119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899
sum0le119904le119886minus1
1 = sum119886119887=119899
119886 = 120590(119899) mentioned inSection 1
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)
References
[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005
[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006
[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg
[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001
[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005
[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001
[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998
[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012
[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995
[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994
[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951
[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004
[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011
[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007
[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010
[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987
[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011
[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012
[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986
[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ
119898timesZ119899rdquo International Journal of NumberTheory
vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat
lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo
Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of