Golestan UniversityFaculty of Sciences

Department of Mathematics

The Extended Abstract of

The 6th National Group TheoryConference

12{13 March 2014

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Invariants of a nite group acted on by a Frobenius-like group . . . . . . . . . . . . . . . . . . . 2Ercan, G.

Representations of a Finite Group with an Extraspecial Normal Subgroup . . . . . . . 3Guloglu, _I. S.

The Classication of Groups via Capability; A Reality to Dream. . . . . . . . . . . . . . . . .9Kayvanfar, S.

Direct Limits of Finitary Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Kuzucuoglu, M.

Algebraically closed groups and embedding theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Shahryari, M.

On the cover-avoiding properties in nite groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26Shum, K. P.

Some open problems in non-commuting graphs of groups . . . . . . . . . . . . . . . . . . . . . . . 28Abdollahi, A.

The relative nth nilpotency degree of two subgroups of a nite group . . . . . . . . . 31Abdul Hamid*, M., Mohd Ali, N. M., Sarmin, N. H. and Erfanian, A.

Generalize commutator on polygroups and hypergroups. . . . . . . . . . . . . . . . . . . . . . . . .35Aghabozorgi*, G. H., Jafarpour, M. and Davvaz, B.

Some solved and unsolved problems in loop theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Ahmadidelir, K.

Inequality for the number of generators of the cnilpotent multiplier . . . . . . . . . . . 46Alizadeh Sanati, M. and Mahdipour*, Z.

Characterization of 2Dn(2) by the set of orders of maximal abelian subgroups . . 50

ii

Asadian*, B. and Ahanjideh, N.

A characterization of Sz(8) by nse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54Asgary*, S. and Ahanjideh, N.

Symmetry classes of polynomials with respect to product of groups . . . . . . . . . . . . . 58Babaei*, E. and Zamani, Y.

(Strongly) Gorenstein homological dimension of groups . . . . . . . . . . . . . . . . . . . . . . . . . 63Bahlekeh, A.

OD-Characterization of the simple group G2(p), where p < 100 . . . . . . . . . . . . . . . . . 67Bibak*, M., Sajjadi*, M. and Rezaeezadeh, G.

Combinatorial conditions on groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Faramarzi Salles*, A. and Khosravi, H.

On the number of elements of a given order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Farrokhi D. G.*, M. and Saeedi, F.

On the lower autocentral series of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Gholamian*, A. and Nasrabadi, M. M.

On a conjecture about automorphisms of nite p-groups . . . . . . . . . . . . . . . . . . . . . . . . 80Ghoraishi, S. M.

Embeddings of borel subgroup of the Ree groups of type 2F4(q2) . . . . . . . . . . . . . . . 84

Ghorbany, M.

The commutativity degree of a polygroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Hokmabadi, A., Mohammadzadeh, F. and Mohammadzadeh*, E.

Finite p-groups whose order of their Schur multiplier is given(t=6). . . . . . . . . . . . . .92Jafari, S. H.

Investigating equality of edge and vertex connectivity number in prime graph ofalternative groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Jahandideh*, M., Kazemi Esfeh, H. and Farhami, N.

iii

A note on the tensor and exterior center of a pair of Lie algebras . . . . . . . . . . . . . . 101Johari*, F., Niroomand, P. and Parvizi, M.

Capability of nite nilpotent groups of class 2 with cyclic Frattini subgroups. . .105Kaheni*, A., Hatamian, R. and Kayvanfar, S.

Support enumerators for some permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Kahkeshani, R. and Yazdany Moghaddam*, M.

On semigroups generated by vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113Khoddami, A. R.

Engel degree and Isoclinism classes of nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Khosravi*, H. and Araskhan, M.

Burnside condition on some intersection subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Mirebrahimi*, H. and Ghanei, F.

Some properties of centralizer and autocommutator subgroup in auto-Engel gro-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Moghaddam, M. R. R. and Badrkhani Asl*, M.

Counting centralizers in non-abelian n-dimensional Lie algebras . . . . . . . . . . . . . . . 127Moghaddam, M. R. R., Hoseini Ravesh, M. and Saarnia*, S.

Some properties of 2-Engel transitive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Moghaddam, M. R. R. and Rostamyari*, A.

Embedding a special subgroup in n-autocentral subgroups of a group . . . . . . . . . . 136Moghaddam, M. R. R. and Sadeghifard*, M. J.

Triangle-free commuting conjugacy classes graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Mohammadian*, A., Erfanian, A. and Farrokhi D. G., M.

Isologism crossed modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Mohammadzadeh, H.

The structure of non-solvable CTI-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Mousavi, H., Rastgoo*, T. and Zenkov, V.

iv

P -semisimple BCI-algebras and adjoint groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Naja*, A. and Rasouli, H.

The nite -solvable groups with three conjugacy class sizes of primary and bipri-mary -elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Naja*, M. and Ahanjideh, N.

On the absolute center of some groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Nasrabadi*, M. M. and Gholamian, A.

On countability of homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165Nasri*, T., Mashayekhy, B. and Mirebrahimi, H.

The schur multiplier of pairs for some nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Nawi*, A. A., Mohd Ali, N. M., Sarmin, N. H., Rashid, S. and Zainal, R.

Separation properties of topological fundamental groups . . . . . . . . . . . . . . . . . . . . . . . 174Pakdaman*, A., Mashayekhi, B. and Torabi, H.

On the characterizations of nite groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .178Parvizi Mosaed*, H., Iranmanesh, A. and Foroudi Ghasemabadi, M.

An approach to c-imperfect groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Pourmirzaei, A. and Hassanzadeh*, M.

The structure of Permutation Groups with t = 1=3(6m 2) . . . . . . . . . . . . . . . . . . . 185Razzaghmaneshi, B.

Permutation Groups with Three Constant Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Razzaghmaneshi, B.

On minimal non PST-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Rezaeezadeh, G. and Aghajari*, Z.

The structure of SS-semipermutable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Rezaeezadeh, G. and Mirdamadi*, S. E.

Movement of permutation groups with two orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199

v

Rezaei, M.

Irreducible characters and conjugacy classes in nite groups . . . . . . . . . . . . . . . . . . . 203Robati, S. M.

The classication of some nilpotent Leibniz 4-algebras . . . . . . . . . . . . . . . . . . . . . . . . . 207Saeedi, F. and Akbarossadat*, S. N.

Finite groups with a given number of relative centralizers . . . . . . . . . . . . . . . . . . . . . .212Saeedi*, F. and Farrokhi D. G., M.

Cellular Automata and its application in group theory. . . . . . . . . . . . . . . . . . . . . . . . .217Safa, H.

On t-extensions of abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Sahleh, H. and Alijani*, A. A.

OD-characterization of almost simple groups related to L2(p2). . . . . . . . . . . . . . . . .224

Sajjadi*, M., Bibak, M. and Rezaeezadeh, G.

A new characterization of Ap where p and p 2 are twin primes . . . . . . . . . . . . . . . 227Salehi Amiri, S. S.

Commuting graphs on conjugacy classes of nite groups . . . . . . . . . . . . . . . . . . . . . . . 230Shadab, M. and Saeidi*, A.

A Quotient OF Topological Fundamental Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Torabi*, H., Pakdaman, A. and Mashayekhy, B.

On 11decomposable nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236Youse*, M. and Ashra, A. R.

The nonabelian tensor square of some nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Zainal*, R., Mohd Ali, N. M., Sarmin, N. H., Rashid, S. and Nawi, A. A.

Epicenter of Lie rings and the Lazard correspondence. . . . . . . . . . . . . . . . . . . . . . . . . .244Zandi, S.

A generalization of Mohres's Theorem on groups with all subnormal sub-

vi

groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Zarrin, M.

* Speaker

vii

Preface

The 6th National Conference on Group Theory was held on the Faculty of Sciencesof Golestan University in Gorgan during 12-13 March 2014. The conference providesa forum for mathematicians and scholar students to present their latest results aboutall aspects of group theory and a means to discuss their recent researches with eachother.

The organizing committee of the conference warmly welcomes the participantsto Gorgan, hoping that their stay here will be happy and fruitful one.

The secretary oce of the conference has received more than 115 papers fromwhich 62 papers have been accepted by the scientic committee.

Al in all, we have made every eort to make the conference as worthwhile aspossible. It is our pleasure to express our thanks to all whose help has made thisgathering possible, particularly the referees of the papers for spending many hoursreviewing papers and providing valuable feedback to the authors; the authors of allsubmitted papers for their contributions and the administration of Golestan Univer-sity.

Chair of Conference: Dr. M. Alizadeh SanatiScientic Chair: Dr. S. M. TaheriExecutive Chair: Dr. A. Pakdaman

viii

General Talks

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

Invariants of a nite group acted on by a Frobenius-like

group

Gul_In Ercan

Middle East Technical University, Ankara, Turkey

ercan@metu.edu.tr

Abstract

Let F be a nite group acted on by a nite groupH via auto-morphisms. This action is said to be Frobenius if CF (h) = 1for all nonidentity elements h 2 H. Accordingly the semidi-rect product FH is called a Frobenius group with kernel Fand complement H whenever F and H are nontrivial. Itis well known that Frobenius actions are coprime actionsand the kernel F is nilpotent. A slight generalization ofthe Frobenius group will be the object of this talk. Moreprecisely, we consider nontrivial nite groups F and H sothat H acts on F via automorphisms, F is nilpotent and[F; h] = F for all nonidentity elements h 2 H, and call thesemidirect product FH a \Frobenius-like group". It shouldbe noted that the group FH is Frobenius-like if and only ifF is a nontrivial nilpotent group and the group FH=F 0 isFrobenius with kernel F=F 0 and complement isomorphic toH:There have been a lot of research about the structure ofnite solvable groups admitting a Frobenius group FH ofautomorphisms. In this talk the action of a Frobenius-likegroup FH on a nite group G will be discussed and con-clusions about some invariants of G and F under additionalhypothesis will be drawn.

2

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

Representations of a Finite Group with an Extraspecial

Normal Subgroup

_Ismail S.Guloglu

Department of Mathematics, Dogus University, _Istanbul,Turkiye

ercan@metu.edu.tr

Abstract

In this talk I will present a technical result which seemsto be of some independent interest and which was obtainedduring our joint research with G.Ercan on the inuence ofxed-point free action of a Frobenius-like group on a solv-able group which will appear in Jour. of Algebra with thetitle \Action of a Frobenius-like Group". We shall prove atheorem about irreducible and faithful , complex representa-tions of a nite group G=PH which has a normal subgroupP isomorphic to an extraspecial group and a complement Hin which each Sylow subgroup is cyclic and H/F(H) is not anontrivial 2-group.

1 Introduction

Many questions about certain invariants of a nite group like nilpotent length, de-rived length, p-length are answered by reducing the relevant group structure to somerelatively simple conguration and then invoking some representation theoretic ar-guments. And most of these representation theoretic arguments are about lineargroups and say that a certain group has regular orbits on the natural module onwhich this linear group acts. Regular modules force the existence of large dimensionand xed points and hence there is a wide range of possible aplications,especially inthe study of groups admitting a xed-point-free group of automorphisms.

The result I want to present in my talk in this conference is a generalization ofa very well known theorem which was proven by Dade, namely the following

Theorem Let H be a group in which each Sylow subgroup is cyclic. Assumethat H=F (H) is not a nontrivial 2-group. Let P be an extraspecial group of order

3

p2m+1 for some prime p not dividing jHj. Suppose that H acts on P in such a waythat H centralizes Z(P ); and [P; h] = P for any nonidentity element h 2 H: Let k bean algebraically closed eld of characteristic not dividing the order of G = PH andlet V be a kG-module on which Z(P ) acts nontrivially and P acts irreducibly. Let be the character of G aorded by V: Then jHj divides pm and H = p

mjHj +

where is the regular character of H, is a linear character of H and 2 f1; 1g.In particular, VH contains the regular kH-module as a direct summand if G is ofodd order.

2 Proof of the Theorem

In this section we present a proof of the theorem.

Lemma 2.1. Let FH be a group with F C FH, F 0 6= F and [F; h] = F for allnonidentity elements h 2 H. Assume that all Sylow subgroups of H are cyclic.Then(i) the groups H 0 and H=H 0 are cyclic of coprime orders,(ii) H = H 0hyi = H0hyi with H 0 \ hyi = 1 for some y 2 H where H0 denotes theFitting subgroup of H, and H0 = H

0 Chyi(H 0) is cyclic,(iii) (H0) = (H).

Proof. The group FH=F 0 is Frobenius with Frobenius complement isomorphic toH. Then (i) follows by [3, Theorem 5.16]. In particular, H = H 0hyi for some y 2 Hwith H 0\hyi = 1. On the other hand the group H has a unique subgroup of order pfor each prime p dividing its order by the argument applied in the proof of Theorem6.19 in [3] which relies on [3, Theorem 6.9]. Hence (H0) = (H) as claimed in(iii). Let now H0 denote the Fitting subgroup of H. Then H0 = H

0(H0 \ hyi) and[H0 \ hyi;H 0] = 1, that is, H0 \ hyi Chyi(H 0) H0. This establishes the claim(ii).

Proof. of The Theorem Since all Sylow subgroups of H are cyclic and G=Z(P )is a Frobenius group with a complement isomorphic to H, we see that H has theproperties described in Lemma 2.1. By [Huppert, V.17.13] we can assume that H isnot nilpotent and recall that H=F (H) is not a 2-group by hypothesis.

Note that dimV = pm as P is a faithful irreducible character of P . Let D be therepresentation of G aorded by the module V and let M be the k-space of squarematrices of size pm over k: We dene a left kH-module structure on M by letting

h X := D(h)XD(h1); for any X 2M and for any h 2 H:

4

It is known that H acts on Homk(V; V ) via the multiplication (h T )(v) = hT (h1v)for any h 2 H, T 2 Homk(V; V ), and v 2 V: Then clearly M is isomorphic to thek[H]-module Homk(V; V ). Furthermore Homk(V; V ) and V

V are isomorphic ask[H]-modules. So by letting Irr(H) = f 1; 2; : : : ; sg and H =

Psi=1 ni i with

nonnegative integers ni; i = 1; : : : s, we have =P

k;l=1;:::;s nknl k l where isthe character of H aorded by M .

Choose a transversal T for Z(P ) in P: Then the set fD(x)jx 2 Tg forms a basisforM by a result of Burnside [Huppert, V.5.14] and the fact that D(zx) = (z)D(x)for any x 2 T and z 2 Z(P ): Notice that P=Z(P ) is the union of one H-orbit oflength 1 and d = p

2m1jHj orbits of length jHj. Thus we haveM = hIiM1 Md

with Mi = k[H] as H-module for any i = 1; 2; : : : ; d: So we get = 1H +

sXi=1

p2m1jHj i (1) i =

sXk;l=1

nknl k l:

Thus the multiplicity of the principal character 1H in is

[1H ;]H = 1 +p2m1jHj =

sXk=1

n2k

and the multiplicity of any nonprincipal 2 Irr(H) in is[;]H =

p2m1jHj (1) =

sXk;l=1

nknl( l; k).

In particular for any nonprincipal linear character of H we havep2m1jHj =

P2Irr(H) nn .

This gives1 =

P2Irr(H) n

2

P2Irr(H) nn , and hence 2 =

P2Irr(H)(n n)2

for any nonprincipal linear character of H.

The group \H=H 0 of characters of the abelian group H=H 0 is isomorphic to H=H 0:In particular it is cyclic. Let # be a generator of \H=H 0. It acts on Irr(H ) bymultiplication. Let i; i = 1; : : : ; b be the orbits of # on Irr(H) and let mi = jij :Then we have 2 =

Pbi=1

P2i(n n#)2. So there are exactly two elements

and in Irr(H) such that jnn#j = 1 = jnn#j; and we have n = n# for any 2 Irr(H)f; g: If 2 i and =2 i, then n 6= n# = n#2 = = n#mi1 =n, which is not possible. So if necessary by reindexing the orbits, we can assumethat and are both elements of b, and n = n# for any i = 1; 2; : : : ; b 1 andany 2 i.

Suppose that = #u for some u 2 f1; 2; : : : ;mb 1g. We haven 6= n# = = n#u 6= n#u+1 = = n#mb1 = n:Since each i is either a #

2-orbit or the union of two #2-orbits of the same size

5

we get

2 =bX

i=1

X2i

(n n#2)2 =X2b

(n n#2)2:

So the dierences n n#2 ; n#mb1 n#; n n#2 are all nonzero ifu 2 f2; : : : ;mb 2g, which is a contradiction. If necessary by replacing # by #1 wecan assume that n 6= n# 6= n#2 = = n#mb1 = n: We let n# = n + ; withsome 2 f1; 1g. Choose an element i from i; i = 1; 2; : : : ; b 1; and let b = :Then

H =

bXi=1

ni(i + i#+ i#mi1) + ; where = b#:

By [Huppert, V.17.13] we have

H0 =

pm 0jH 0j

0 + 00 = (bX

i=1

nimii)H0 + bH0

for some 0 2 f1; 1g and 0 2 Irr(H 0) where 0 is the regular character of H 0.It follows by that if i 6= j then the sets of irreducible constituents of the restric-

tions of i and j are disjoint. By Cliord's theorem we have

iH0 = eitiPj=1

i;j where IH(i;1) = Ti, ti = [H : Ti], H =tiSj=1

Tixi;j ; and i;j =

xi;ji;1 ; j = 1; 2; : : : ; ti; i = 1; 2; : : : ; b: Now fi;j jj = 1; 2; : : : ; ti; i = 1; 2; : : : ; bg =

Irr(H 0).It is known that there is a unique i 2 Irr(Ti) such that iH = i and iH0 =

eii;1: On the other hand as Ti=H0 is cyclic, i;1 has an extension, say ', to Ti. But

then 'H must belong to the #-orbit containing i which implies iH0 = ('H)

H0 .Therefore we have

ei = [iH0 ; i;1] = [('H)

H0 ; i;1] = ['H0 ; i;1] = 1 for any i = 1; 2; : : : ; b:

Let now e = pm0jH0j and

0 = i0;j0 . Then for any 2 cH 0 we have[

H0 ; ]H0=

e if 6= 0

e+ 0 if = 0 :

Set H0 = F (H): Applying [Huppert, V.17.13] to the action of PH0 on V we seein particular that jH0j divides pm for some 2 f1; 1g. Then jH 0j dividesa^e2 = (pm ) (pm a^e2) and so we have either a^e2 = or jH 0j = 2.If the latter holds then H 0 Z(H) and hence H is abelian, which is not the case.Thus jH0=H 0j divides e. In particular e > 1 and so e + 0 > 0 which shows that[H ; i0 ]H 6= 0:

6

If ti0 6= 1; then there exists j1 6= j0 such thate = [

H0 ; i0;j1 ]H0 = [H0 ; i0;j0 ]H0 = e+ 0

which is not possible. Then ti0 = 1 and hence 0 is H-invariant. This yields that

i0H0= 0 = i0;1. In particular i0 is a linear character of H and so mi0 = jH=H 0j.

Furthermore we have

e+ 0 =

ni0mi0 if i0 < b

nbmb + if i0 = b:

Now jH0=H 0j divides the greatest common divisor of e and mi0 which forces thati0 = b as H0=H

0 is nontrivial. Furthermore if 6= 0 we have jH0=H 0j = 2, whichimplies by Lemma 2.1 that H=H 0 is a 2-group. This contradiction shows that = 0

and hence nbmb = e by the above formula. In particularpmjHj = nb is an integer.

On the other hand we also have e = [H0 ; i;1]H0 = nimi if i < b.

Set next ri = jTi=H 0j : As \Ti=H 0 = h#jTii we obtain Ti Ker#ri : As i = iHfor some i of Ti and Ti is normal in H, we observe that i(x) = 0 for any x =2Ti. Combining these two observations we get #

rii = i: Thus mi divides riand hence jH=H 0j = riti = miciti for some positive integer ci: It follows now thatnimiciti = ecii(1) and hence ni =

pmjHj cii(1) p

mjHj i(1): Thus

pmjHj +

occurs in H . As the degrees of these characters are the same we see that they areequal. This completes the proof of the theorem.

The next example shows that the hypothesis about the structure of H can notbe avoided.Example Let V be the GF (3)-space GF (34): We dene the map(j) : V V ! GF (3) by (j)(x; y) = Tr(d (xy9 x9y)) for x; y 2 V , whered is an element of order 16 in GF (34): One can check that (j) is a nonsingularsymplectic form on V:

Let b 2 GF (34) be an element of order 5 and c 2 GF (34) be an element oforder 4. We dene b : V ! V by b(x) = b x and : V ! V by (x) = c x9:Then H = hb; i is a subgroup of GL(4; 3) preserving the symplectic form, withjHj = 20;H 0 = hbi of order 5, and F (H) = H 0 h2i of order 10: Furthermoreh(v) = v for some 0 6= v 2 V and h 2 H implies that h = 1. So if P is theextraspecial group of order 35 and exponent 3; then it admits H as a subgroup ofautomorphisms of P; centralizing Z(P ) and satisfying [P; h] = P for any nonidentityelement h 2 H: Let be any irreducible character of the group PH which doesnot contain Z(P ) in its kernel. Clearly, we have H 6= 3

2jHj + for the regular

H-character and any 2 f1; 1g and 2 Irr(H), because 32jHj is not an integer.

7

References

[1] B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin-New York 1967.

[2] G. Ercan, _I. S. Guloglu: "Action of a Frobenius-like Group" 2014, will appearin J. Algebra.

8

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

The Classication of Groups via Capability; A Reality

to Dream

Saeed Kayvanfar

Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

skayvanf@math.um.ac.ir; skayvanf@yahoo.com

Abstract

This talk is a survey article on the classication of groups.The classication of prime power groups of order at most p6

was done using the notion of isoclinism and invoking a fun-damental instrument, namely the capability of groups. Sincethe basic concepts of this classication i.e., isoclinism andcapability were generalized to any variety of groups, there-fore this talk intends to propound a basic question whetherit is possible to dene some suitable varieties that could playthe key role for classifying some other families of groups.

1 Introduction

One of the most important problem in group theory is the classication of groups.The problem which has been always studied along with the age of group theory.There have been also various approaches to face the problem. Among several dif-ferent approaches, one of the most classical notions is the concept of isomorphismbetween groups. However, this notion is too strong in many cases. For this reason,P. Hall in 1940 [5] introduced the notion of isoclinism between two groups (whichis weaker than isomorphism) and could classify some groups of prime power order.Using his method, his student Eastereld [6] and then M. Hall and J. Senior in 1964[6] and later R. James in 1988 [8] completed the classication of groups of order atmost pn, where p is a prime and n is at most 6.

2010 Mathematics Subject Classication. Primary 20E10; Secondary 20D15.Key words and phrases. Classication, isologism, capability.

9

We also know that the notion of isoclinism was generalized by P. Hall [5] toisologism. Isologism is in fact isoclinism with respect to a certain variety of groups.If one takes the variety of all trivial groups, one gets the notion of isomorphism back.The variety of all abelian groups yields isoclinism.

On the other hand, we know that the main strong tool in the P. Hall's classica-tion is the notion of capability and this notion also was simultaneously generalizedto varietal capability by J. Burns and G. Ellis [3] and a joint paper of the author [9]in 1997.

Now, using all these facts in hand, we just intend to propound a fundamentalquestion; Is it possible to dene some suitable varieties so that invoking them theclassication of some other families of groups happens? Indeed, in the face of thesefacts, it seems that the classication of some other suitable families of groups will notbe so far! By this sentence, we mean that it might probably exist some special kindof varieties such that their obtained classes caused by isologim can be consideredas the rst step of screening in the classication, though they might be so broad.Although we do understand that as usual, the stating such a problem is so easywhile nding the answer may not!

2 Reality; The Classication of some Prime Power Groups

There are dierent approaches that can be considered for the description of nitep-groups. We used the word \description" rather than \classication" because weknow that classifying p-groups is notoriously open and dicult. Some of the ap-proaches are, for instance, the order, the coclass and the class of nilpotency. Buteach of them has some restrictions and diculties in this way so that P. Hall in 1940[5] introduced the notion of isoclinism for the classication of all groups, though hecould classify only some prime power groups.

Denition 2.1. Two groups G1 and G2 are said to be isoclinic provided that thereexist two isomorphisms : G1=Z(G1) ! G2=Z(G2) and : 2(G1) ! 2(G2)such that if (a1Z(G1)) = a2Z(G2) and (b1Z(G1)) = b2Z(G2), then ([a1; b1]) =[a2; b2]: This notion is written by G1 G2:

In the P. Hall's classication, regular group which was dened by him, plays thekey role. There are many equivalent ways to dene regular p-groups. One is that ifa and b are any elements of the group, then

(ab)pr= ap

rbp

rc1pr : : : cp

r

t ;

where ci are elements of the commutator subgroup of < a; b >. In fact, a group isregular if the operation of taking pth powers interacts \well" with taking commuta-tors. P. Hall [5] showed that in a regular p-group, one can dene \type invariants"

10

which are similar to the invariant factors for nite abelian groups. Though they donot completely determine the groups the way the invariant factors do for abeliangroups, they are usually a very good rst reduction towards the analysis. Note thatif p n, then a group of order pn is necessarily regular (more generally, if the groupis of class c and p > c, then the group is regular, in particular, since a group oforder pn is of class at most n 1, the observation just made follows). In fact, P.Hall mentioned in [4] that if we x n, then \most" groups of order pn are regular(since only those with p < n may fail to be regular). This leads, classically, to aseparation of p-groups into those of \small class" (when the class is smaller than p),and \the rest". In other words, this means that when classifying groups of orderpn, the analysis usually breaks into two dierent cases: when the group is regular(which includes all p n), and when the group is irregular. The latter case leadsto a case-by-case analysis for small primes. This occurs, for example, in the clas-sication of groups of order p3 (in which odd primes and 2 should be consideredseparately). Likewise Burnside's work on group of order p4. Or similar to the workof R. James [8] in the classication of groups of order p6 and E. A. O'Brien and M.R. Vaughan-Lee [10] for p7. The latter separates p 7 with the groups of order 37and 57. Note that the latest work uses Lie rings and algebras as a starting point.There are algorithms that are known to produce and check isomorphism types (see[10]).

The above comments explains that how groups of order pn for all primes p havebeen fully classied for n 7, rst (most of the times) by isoclinism and then up toisomorphism. Note, in particular, that the number of isomorphism classes increaseswith p when n 5. One may know that the Higman PORC conjecture (polynomialon residue classes) is that, for each n, this number is a polynomial function of p andof p (mod k) for a nite collection of values of k. Although a family of examplesconstructed by M. D. Sautoy and M. R. Vaughan-Lee [12] of order p10 show whilenot actually disproving the conjecture, suggests that it is very unlikely indeed thatit is true. We should remind also that their construction depends on the geometryof elliptic curves. It will probably illustrate that how describing all groups of orderp10 would be complicated.

Let us comeback to the isoclinism. The notion of isoclinism denes an equivalencerelation on the class of all groups and has this trait that some other propertiesof groups are invariant under isoclinism. For instance, it is proved in [2] that,restricting ourselves to nite groups, we have the following hierarchy of classes ofgroups, invariant under isoclinism: abelian < nilpotent < supersolvable < stronglymonomial < monomial < solvable. For charactering the families of isoclinism, P.Hall [5] tried to nd some properties which are invariant in each family. Accordingly,any quantity depending on a variable group and which is the same for any two groups

11

of the same family is called a family invariant. For instance, it is easy to see that themembers of the derived series and the central quotient groups are family invariants.It follows that the groups belonging to the same family have the same derived lengthand nilpotency class. Note that the commutator quotient group and the center arenot family invariants, as is the minimal number of generators.

Now, the importance of central quotient groups for this classication may beseen. Such groups are called capable. More precisely;

Denition 2.2. A group G is called capable if there exists a group E such thatG = EZ(E) :

Among many dierent points on capability, there are two important tools forrecognition the capability of a group. One of them is a necessary condition whichwas established by P. Hall [5]. He considered a generating system J for a group Gand dened J = \x2J < x > and denoted the join of all subgroups J , where Jvaries over all generating systems of G, by (G) and gave a necessary condition forthe capability of G in such a way that a capable group G must satisfy (G) = 1.

The other tool for characterizing the capability is a criterion that was introducedby F. R. Beyl, U. Felgner and P. Schmid [1]. They showed that every group Gpossesses a uniquely determined central subgroup Z(G) which is minimal subjectto being the image in G of the center of some central extension of G. This Z(G) ischaracteristic in G and is the image of the center of every stem cover of G.

Denition 2.3. The intersection of all subgroups of the form (Z(E)), where :E ! G is a surjective homomorphism with ker Z(E); is called the precise centersubgroup of G and denoted by Z(G).

F. R. Beyl et al. [1] proved that Z(G) is the smallest central subgroup of Gwhose factor group is capable. Now using this fact the criterion for capability canbe given as follows.

Theorem 2.4. (F. R. Beyl, U. Felgner and P. Schmid 1979 [1]) G is capable if andonly if Z(G) = 1.

3 Dream; The Classication of Other Classes of Groups!

The works and attempts which were explained as the \Reality" in Section 1 for theclassication of some prime power groups evidence that this way of classicationmight not be quite simple. In other words, determining nite p-groups of order pn

up to isomorphism, will be so complicated while n becomes large and larger. Thisshows the necessity of primary screening not only for p-groups, but also for otherfamilies of groups.

12

On the other hand, we should not forget that whatever, for instance, R. James[8] did was heavily indebted to the work of P. Hall [5] on isoclinism. That is why,we believe that as isoclinism could help us as the rst step of classication of primepower groups of order pn for small n, paying attention to its generalization may behelpful as the rst step of classication of some other classes of groups! Therefore inthis section, we remind all the notions and the tools in Section 1 which were gener-alized to any variety of groups. In fact, the primary denitions and the preliminarystatements which provide the context of determining of the equivalence classes inisoclinism were generalized in two steps. First, the \isoclinism" and \capability"were transformed to \c-isoclinism" and \ccapability and then they were gener-alized to \isologism" and \varietal capability", respectively. In the following, thegeneral case is provided.

Denition 3.1. Let V be a variety of groups dened by the set of laws V . Twogroups G and H are V- isologic if there exist isomorphisms

:G

V (G)! H

V (H)and : V (G)! V (H);

such that (v(g1; g2; : : : ; gn)) = v(h1; h2; : : : ; hn); where gi 2 G; hi 2 (giV (G))for each 1 i n: In this case, we write G V H: The pair (; ) is said to be aV-isologism between G and H.

Likewise the isoclinism, for each varietyV, isologism gives an equivalence relationon the class of all groups. The larger variety implies the weaker equivalence relation.If V is the variety of all abelian groups, V-isologism coincides with isoclinism. Thegroups in a variety V fall into one single equivalence class, they are actually V-isologic to the trivial group. (For more information about V-isologism see [7]).

On the other hand, it is observed from the denition of isologism that themarginal factor group can play an important role in this system of classication.Such a group is called varietal capable with respect to the variety V, or brieyV-capable [9].

Denition 3.2. Let V be a variety of groups dened by the set of laws V . A groupG is said to be V-capable , if there exists a group E such that G = E=V (E) .

Some properties of varietal capability are given in [9]. Specially, nding a cri-terion for recognition of varietal capability is illustrated in [9]. More precisely, it isshown that every group G possesses a uniquely determined subgroup (V )(G) ofthe marginal subgroup V (G) , which is minimal subject to being the image in G ofthe marginal subgroup of some V-marginal extension of G. In fact, if : E ! G isa surjective homomorphism with ker V (E), then (V )(G) is dened to be theintersection of all subgroups of the form (V (E)).

13

If V is the variety of abelian groups then the subgroup (V )(G) is Z(G) andthe V-capability coincides with the usual capability. If one takes V to be the varietyof nilpotent groups of class at most c, c 1, then one gets (V )(G) to be Zc (G) asJ. Burns and G. Ellis introduced in [3] and the V-capability will be their c-capability.

(V )(G) is characteristic and it is also proved in [9] that (V )(G) is the small-est subgroup contained in the marginal subgroup of G for which the factor groupG=(V )(G) is V-capable. In other words;

Theorem 3.3. (M. R. R. Moghaddam and S. Kayvanfar [9]) G is V-capable if andonly if (V )(G) = 1.

The above comments explain that for a classication of a family of groups, thenotion of isologism might be helpful as a primary screening. The most importanttool that can be considered for characterizing the families of isologism is the V-capable group. On the other hand, there are a few statements for recognition thevarietal capability (for instances, see [3] and [9]) invoking them one can nd outthe V-capability of some groups. There is also another helpful tool for recognizingthe V-capability; the Baer invariant of groups, which has been calculated for manyvarieties. Invoking the Baer invariant, the varietal capability of some types of groupshas been characterized for some special kind of varieties (for example see [11]).

All these facts motivate us to think more to the dream of classication of groupsby varietal isologism via the varietal capability.

Acknowledgement

The author would like to thank Dr. A. Kaheni with whom I have had many usefulconversations on the classication of groups.

References

[1] F. R. Beyl, U. Felgner and P. Schmid, On groups occuring as centre factorgroups, J. Algebra 61 (1979), 161{177.

[2] J. C. Bioch and R. W. van der Waall, Monomiality and isoclinism of groups, J.reine ang. Math. 298 (1978), 74{88.

[3] J. Burns and G. Ellis, On the nilpotent multipliers of a group, Math. Z. 226(1997), 405{28.

[4] P. Hall, A contribution to the theory of groups of prime power order, Proc.London Math. Soc. (series 2) 36 (1934), no.1, 29{95.

14

[5] P. Hall, The classication of prime-power groups, J. reine ang. Math.182 (1940),130{141.

[6] M. Hall, Jr., and J. K. Senior, The groups of order 2n(n 6), Macmillan, NewYork, 1964.

[7] N. S. Hekster, Varieties of groups and isologism, J. Austral. Math. Soc. (seriesA) 46 (1989), 22{60.

[8] R. James, The groups of order p6 (p an odd prime), Math. of Computation 34no. 150 (1980), 613{637.

[9] M. R. R. Moghaddam and S. Kayvanfar, A new notion derived from varietiesof groups, Algebra Colloquium 4:1 (1997), 1{11.

[10] E. A. O'Brien and M. R. Vaughan-Lee, The groups with order p7 for odd primep, J. Algebra 292 (2005), 243{258.

[11] M. Parvizi, B. Mashayekhy and S. Kayvanfar, Polynilpotent capability of nitelygenerated abelian groups, Journal of Advanced Research in Pure Mathematics2(3) (2010), 81 { 86.

[12] M. D. Sautoy and M. R. Vaughan-Lee, Non-PORC behaviour of a class ofdescendant p-groups, arXiv:1106.5530, 30 Jan, 2013.

15

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

Direct Limits of Finitary Symmetric Groups

Mahmut Kuzucuoglu

Middle East Technical University Department of Mathematics Ankara, Turkey

matmah@metu.edu.tr

Abstract

We describe the construction of a new class of simple locallynite groups as a direct limit of nitary symmetric groups.Moreover we investigate the structure of the centralizers ofelements in these groups.

1 Introduction

A group is called a locally nite group if every nitely generated subgroup is anite group. One of the natural construction of locally nite groups is by taking thedirect limit of nite groups. Using this method uncountably many, simple locallynite groups of countably innite order is constructed as a direct limit of nitesymmetric groups in Kegel-Wehfritz [2, Chapter 6], (1976). Then the classication ofthese groups by using Steinitz numbers is done by N. V. Kroshko-V. I. Sushchanskyin [3], (1998). We now describe this construction.

Let be the permutation dened by =

1 : : : n

i1 : : : in

. Then the permutation

dr() =

1 : : : n n + 1 : : : 2n : : : (r 1)n + 1 : : : rni1 : : : in n + i1 : : : n + in : : : (r 1)n + i1 : : : (r 1)n + in

is called a homogeneous r-spreading of the permutation .Let be the set of sequences consisting of prime numbers and 2 . So =

(p1; p2; : : :) is a sequence consisting of not necessarily distinct primes pi. We obtaindirect systems by using homogeneous pi-spreading from the following embeddings

f1g dp1! Sn1 dp2! Sn2 d

p3! Sn3 dp4! : : :

2010 Mathematics Subject Classication. Primary:20F50; Secondary: 20E32.Key words and phrases. Centralizers, Direct Limit groups, Simple groups.

16

f1g dp1! An1 dp2! An2 d

p3! An3 dp4! : : :

where n0 = 1, n1 = p1; ni = ni1pi; i = 2; 3 : : : and Sni is the symmetric groupon ni letters, Ani is the alternating group on ni letters. The direct limit groupsobtained from the above direct systems are of strictly diagonal type and denoted byS() and A(); respectively. Observe that S() Sym(N).

It is proved that such groups satisfy the followings:

If the prime 2 appears innitely often in the sequence , then the limit groupS() is a simple non-linear locally nite group.

If a prime p appears innitely often, then S() contains an isomorphic copy ofthe locally cyclic p-group Cp1 .

For each sequence , we dene Char() = pr11 pr22 : : : where ri is the number oftimes that prime pi repeat in . If it repeats innitely often, then we writep1i . Therefore for each there corresponds a Steinitz number Char(). For agroup S() obtained from the sequence we dene Char(S()) = Char()

Two groups S(1) and S(2) are isomorphic if and only if Char(S(1)) =Char(S(2)).

There are uncountably many non-isomorphic simple locally nite groups ofthis type.

We will discuss some of the results for the centralizers of elements in S(), inparticular the following Theorem.

Theorem 1.1. (Guven, Kegel, Kuzucuoglu [1]) Let be an innite sequence, g 2S() and the type of principal beginning g0 2 Snk be t(g0) = (r1; r2; : : : ; rnk): Then

CS()(g) =nkDri=1

Ci(Cio S(i))

where Char(i) =Char()

nkri for i = 1; : : : ; nk: If ri = 0; then we assume that

corresponding factor is f1g:Let be an arbitrary set and Sym() be the symmetric group on the set .

Let g 2 Sym(). We dene Supp(g) = f 2 j g() 6= g the set of el-ements of which are moved by the permutation g. Then FSym() = fg 2Sym()j jSupp(g)j

be the set of sequences of prime numbers and 2 . Then is a sequence of notnecessarily distinct primes. Let 2 FSym(), ( Alt() ). For a natural numberp 2 N a permutation dp() 2 FSym(p) dened by (s+ i)dp() = s+ i; i 2 and 0 s p 1 is called a homogeneous p-spreading of the permutation .We divide the ordinal p into p equal parts and on each part we repeat the permu-

tation diagonally as in the nite case. So if =

1 : : : n

i1 : : : in

2 FSym(); then the

homogeneous pspreading of the permutation is

dp() =

1 : : : n + 1 : : : + n : : : (p 1) + 1 : : : (p 1) + ni1 : : : in + i1 : : : + in : : : (p 1) + i1 : : : (p 1) + in

with the assumption that the elements in p n supp(dp()) are xed.We continue to take the embeddings using homogeneous p-spreadings with re-

spect to the given sequence of primes in . From the given sequence of embeddings,we have direct systems and hence direct limit groups FSym()(); (Alt()()). Ob-serve that FSym()() and Alt()() are subgroups of Sym(!):

The principal beginning 0 of an element 2 FSym()() is dened to be thesmallest positive integer nj 2 N such that 0 2 FSym(nj) and 0 is not obtainedas a sequence of embeddings dpi for any pi 2 .

Theorem 1.2. (Guven, Kegel, Kuzucuoglu [1]) Let be an innite sequence. If 2 FSym()() with principal beginning 0 2 FSym(ni); t(0) = (r1; : : : ; rn);and jsupp(0)j = n. Then

CFSym()()() =n

(Dri=1

Ci(Ci o S(i))) FSym()(0)

where Char(i) =Char()

niri and Char(

0) = Char()ni . If ri = 0, then we assume thatthe corresponding factor in the direct product is f1g.

References

[1] Guven U. B., Kegel O. H., Kuzucuoglu M.; Centralizers of subgroups in di-rect limits of symmetric groups with strictly diagonal embedding, To appear inCommunications in Algebra .

[2] Kegel O. H., Wehrfritz B. A. F.; Locally Finite Groups, North-Holland Publish-ing Company - Amsterdam, 1973.

18

[3] Kroshko N. V.; Sushchansky V. I.; Direct Limits of symmetric and alternatinggroups with strictly diagonal embeddings, Arch. Math. 71, 173{182, (1998).

19

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

Algebraically closed groups and embedding theorems

M. Shahryari

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz,Tabriz, Iran

mshahryari@tabrizu.ac.ir

Abstract

Using the notion of algebraically closed structures, we obtainnew embedding theorems for groups and Lie algebras. Wealso prove the existence of some groups and Lie algebraswith prescribed properties.

The group Z2 is the only nite group which has just two conjugacy classes. Is thereany innite group with the same property? Denis Osin [7], proved that there is anitely generated innite group with exactly two conjugacy classes. His method isbased on the small cancellation theory over relatively hyperbolic groups. Anotherexample of such groups is obtained by Higman, Neumann and Neumann using theirwell-known embedding methods, [3]. In this article, we are dealing with problemslike this. Using the concept of algebraically closed groups, we prove that any R-groupcan be embedded in a Q-group which has only two conjugacy classes. Rememberthat a group G is called R-group, if for any integer m, the equality xm = ym impliesx = y. We also give a generalization of these groups and dene an R-group tobe a group G such that the equality xm = ym implies x = y, whenever m is a 0-number. We prove that any R-group can be embedded in a simple Q-group, whoseelements of the same order are conjugate. We study many interesting properties ofsuch groups and then we obtain similar results for Lie algebras. By the notion ofalgebraically closed Lie algebras, we show that any Lie algebra L can be embeddedin a simple Lie algebra L which has many rare properties; for any non-zero elementsa and b, there is x such that [x; a] = b, and so for all non-zero x the derivation ad xis not nilpotent. In the case of nite elds, every nite dimensional Lie algebra can

2010 Mathematics Subject Classication. Primary 20E45, Secondary 20E06.Key words and phrases. Conjugacy classes, R-groups; R-groups, algebraically closed groups;

HNN-extension; Q-groups; Lie algebras; derivations; embedding theorems, Monsters

20

be embedded in L and it is possible to describe the derivation algebra of any nitedimensional algebra A as the quotient algebra NG(A)=CG(A). We also prove theexistence of some groups and Lie algebras with prescribed properties. Our maintool in this work is HNN-extensions of groups and Lie algebras. The rst one iswell-known and the reader can consult any book on combinatorial group theory(for example [3]) to see denition and properties of HNN-extensions of groups. TheHNN-extensions of Lie algebras are not so popular and it seems that there are onlytwo articles ever published in the subject, [2] and [11].

Although, we are dealing just with groups and Lie algebras in this article, it isuseful to give a general denition of algebraically closed structures in the frames ofthe universal algebra. Let L be an algebraic language and A be an algebra of typeL. We extend L to a new language LA by adding new constant symbols ca for anya 2 A. Let TA(X) be the term algebra of LA with variables from a countable set X.If p(x1; : : : ; xn) and q(x1; : : : ; xn) are elements of this term algebra, then we call theexpression p(x1; : : : ; xn) = q(x1; : : : ; xn) an equation with coecients from A. Aninequation is the negation of an equation. A system of equations and inequationsover A (or a system over A) is a nite set consisting equations and inequations. Wesay that A is algebraically closed (a.c. for short), if and only if any system over Ahaving a solution in an extension B of A, has already a solution in A. In this article,we consider the case of groups and Lie algebras, but some of the theory here, can begeneralized to arbitrary algebraic structures. Note that in the case of a group G, wemay assume that an equation has the form w(x1; : : : ; xn) = 1, where w is an elementof the free product G F (X). Here F (X) is the free group on the set X. Similarly,if L is a Lie algebra (or any non-associative algebra), then an equation over L hasthe form w(x1; : : : ; xn) = 0, with w and element of the free product LF (X), whereF (X) is the free Lie algebra over X.

1 Algebraically closed groups

A group G is called algebraically closed, if any nite consistent system of equationsand inequations with coecients from G has a solution in G. A system

S = fwi(x1; : : : ; xn) = 1; (1 i r); wj(x1; : : : ; xn) 6= 1; (r + 1 j s)g (I)

with coecients in G is called consistent, if there is a group K containing G, suchthat S has a solution in K. One can generalize this denition to an arbitrary classof groups: Let X be a class of groups. A group G 2 X is called algebraically closed inthe class X, if every X-consistent system S has a solution in G. Here, X-consistencymeans that there exists a group K 2 X which contains G and S has a solution in K.

21

Recall that a class of groups is called inductive, if it contains the union of anychain its elements. We will use the next theorem to obtain our embedding resultson groups.

Theorem 1.1. Let X be an inductive class of groups which is closed under theoperation of taking subgroups. Let G 2 X. Then, there exists a group G 2 X, withthe following properties,

1- G is a subgroup of G.2- G is algebraically closed in the class X.3- jGj maxf@0; jGjg.We use the concept of algebraically closed groups and above theorem to embed

R-groups in Q-groups of the same cardinality with just two conjugacy classes. Then,we will extend our results to the class of R-groups for any set of primes . Ourmain tool in this section is HNN-extensions of groups and normal forms of elementsin such groups, [3]. Recall that an R-group is group for which xm = ym impliesx = y for any non-zero integer m. A divisible R-group is a rational exponentialgroup or a Q-group in other words. The reader can consult [4] or [5] for a theory ofexponential groups.

Theorem 1.2. Let G be an R-group. Then G can be embedded in a Q-group Gwith only two conjugacy classes. Further jGj = jGj.

We can generalize the embedding theorem of R-groups to a more general classesof groups. Let be a set of prime numbers. We consider the ring

Q = fmn2 Q : n is a 0 numberg:

If consists of a single element p, then we denote this ring by Qp. A group G is anR-group, i for any x and y and any

0-number n, we have the implication

xn = yn ) x = y:

The order of any element of such a group is innite or a -number. We say that Gis -divisible, i for any x 2 G and any 0-number n, there exists y 2 G, such thatyn = x. Note that if a group G is both R-group and -divisible, then there is aunique y satisfying yn = x, for a given 0-number n. So, we denote this element byx

1n . Now, a group which is both R-group and -divisible, can be regarded as an

exponential group over the ring Q (or Q-group for short) via xmn = (x

1n )m. The

reader most consult [4] and [5] for the theory of exponential groups. Note that anyQ-group is also R-group and -divisible.

22

Theorem 1.3. Let G be an R-group. Then there exists a Q-group G containingG and with same cardinality as G, such that

1- G is simple,2- element of the same order in G are conjugate,3- G is not nitely generated,4- every nite -group embeds in G.5- every nitely presented -group can be residually embedded in G,6- for any nite -group A, we have Aut(A) = NG (A)CG (A) .

2 Algebraically closed algebras

Our next mission is to nd similar embedding theorems for Lie algebras. But thereare two major dierences between groups and Lie algebras. First, we don't haveany suitable denition of torsion in the case of Lie algebras so, in advance, we don'thave a parallel concept of R-Lie algebra and so on. Instead, we can express ourtheorems in terms of arbitrary Lie algebras. The second main dierence is relatedto HNN-extensions of Lie algebras. Here, an HNN-extension comes from a Liealgebra and a derivation of some subalgebra, despite groups where HNN-extensionsare always dened by groups and isomorphisms between subgroups. We will givea brief summary of HNN-extensions of Lie algebras in the next section. In thissection, we give the analogue of Theorem 1.1 for Lie algebras, in fact since it can beformulated for arbitrary non-associative algebras, we prove it in the most generalform.

Theorem 2.1. Let X be an inductive class of (not necessarily associative) algebrasover a eld K. Suppose X is closed under subalgebra and L 2 X. Then there existsan algebra L 2 X with the following properties,

1- L is a subalgebra of L.2- L is algebraically closed in the class X.3- dimL maxf@0; dimL; jKjg.In [2] and [11], the concept of the HNN-extension is dened for Lie algebras.

Suppose L is a Lie algebra over a led K and A be a subalgebra. Let : A! L bea derivation. Dene a Lie algebra L with the presentation

L = hL; t : [t; a] = (a); (a 2 A)i:The properties of this HNN-extension is studied in [2] and [11]. It is proved that Lis a subalgebra of L. Similar constructions are also introduced for Lie p-algebrasand rings in [2]. In this section, using this HNN-extension, the theorem above, andthe notion of algebraically closed Lie algebras, we obtain a new embedding theorem.

23

Theorem 2.2. Let L be a Lie algebra over a eldK. Then there exists a Lie algebraL having the following properties,

1- L is a subalgebra of L,2- for any non-zero a; b 2 L, there exists x 2 L such that [x; a] = b,3- for any non-zero x 2 L the derivation ad x is never nilpotent,4- L is simple,5- dimL maxf@0; dimL; jKjg,6- L is not nitely generated,7- every nite dimensional simple Lie algebra over K embeds in L,8- every nitely presented Lie algebra over K embeds residually in L,9- if K is nite, then every nite dimensional Lie algebra over K embeds in L,10- if K is nite and A is nite dimensional Lie algebra over K, then we have

Der(A) = NL(A)CL(A)

:

3 Some Olshanskii like groups

In mid twenties, Alfred Tarski asked about the existence of innite groups all propernon-trivial subgroups of which are of xed prime order p. In 1982, A. Yu. Olshanskii[6], constructed an uncountable family of such groups using his geometric method ofgraded diagrams over groups, for all primes p > 1075. The groups constructed arecalled Tarskii monsters since then. These groups are two-generator simple groupsand hence are countable. In this section, for any xed prime p, we give a quite ele-mentary proof for existence of countable non-abelian simple groups with the propertythat their all non-trivial nite subgroups are cyclic of order p.

We will consider two special classes of groups in this section. The rst oneconsists of groups all nite subgroups in which are cyclic. We will denote this classby Xfc. The second class which will be denoted by Xp, is the class of all groupsin which their non-trivial nite subgroups are of order p, for a xed prime p. Notethat both classes are inductive and closed under subgroup. Clearly the Monstersconstructed by Olshanskii satisfy the requirements of the next theorem, but we don'tuse that monsters, since we have a very elementary proof for our claims. What weneed is the theorem 1.2 and some facts about nite subgroups of HNN-extensions(and also those of free products). It is known that (see [3], page 212) every nitesubgroup of any HNN-extension

G = hA; t : tF t1 = (F )iis contained in some conjugate of A. Also, every nite subgroup of any free productA B is contained in some conjugate of A or some conjugate of B.

24

Theorem 3.1. There exists a countable non-abelian simple group M such that allnite subgroups of M are cyclic and for any prime p, the group M has an elementof order p.

A similar result can be obtained if we use the class Xp.

Theorem 3.2. Let p be a xed prime. Then there exists a countable non-abeliansimple group M (which is not torsion free) such that any nite non-trivial subgroupof M is cyclic of order p.

References

[1] Higman, G., Scott, E. L. Existentially closed groups, Clarendon Press, 1988.

[2] Lichtman, A. I., Shirvani, M. HNN-extensions for Lie algebras, Proc. AMS, Vol.125, No. 12, pp. 3501-3508, 1997.

[3] Lyndon, R. C., Schupp, P. E. Combinatorial group theory, Springer-Verlag,2001.

[4] Myasnikov A. G., Remeslennikov V. N. Exponential groups I: fundations of thetheory and tensor completions, Siberian Math. J. Vol 35, No. 5, pp. 986-996,1994.

[5] Myasnikov A. G., Remeslennikov V. N. Exponential groups II: extensions ofcentralizers and tensor completion of CSA-groups, International J. Algebra andComputation, Vol. 6, No. 6, pp. 687-711, 1996.

[6] Olshanskii, A. Y. Geometry of dening relations in groups, Kluwer AcademicPublishers, 1991.

[7] Osin, D. Small cancellations over relatively hyperbolic groups and embeddingtheorems, Ann. of Math. (2), 172 (1), pp. 1-39, 2010.

[8] Scott, W. R. Algebraically closed groups, Proc. of AMS, No. 2, pp. 118-121,1951.

[9] Shahryari, M. A note on derivations of Lie algebras, Bull. Aust. Math. Soc.Vol. 84, pp. 444-446, 2011.

[10] Shahryari, M. Embeddings coming from algebraically closed groups, submitted.

[11] Wasserman, A. A derivation HNN construction for Lie algebras , Israil J. Math.Vol. 106, pp. 76-92, 1998.

25

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

On the cover-avoiding properties in nite groups

Kar Ping Shum

The Chinese University of Hong Kong

.

Abstract

In this talk, I will talk about the process of the investigationon cover-avoiding properties. Mainly talk about:

Cover-avoiding properties and the structure of nitegroups

Semi cover-avoiding properties and the structure of -nite groups

Further investigationsAll groups mentioned here are nite.

26

Talks

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

Some open problems in non-commuting graphs of

groups

Alireza Abdollahi

Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iranand School of Mathematics, Institute for Research in Fundamental Sciences (IPM),

P.O.Box: 19395-5746, Tehran, Iran

a.abdollahi@math.ui.ac.ir

Abstract

Let G be a non-abelian group. The non-commuting graph ofG, denoted by G, is the graph whose vertex set is G and twovertices are adjacent if they do not commute. In this talk,we briey review some open problems about non-commutinggraphs of nite groups.

1 Introduction

Let G be a non-abelian group. The non-commuting graph of G, denoted by G,is the graph whose vertex set is G and two vertices are adjacent if they do notcommute. The non-commuting graph is studied in [2]. Here we briey review someopen problems about non-commuting graphs of nite groups.

2 Order Conjecture

It is conjectured in [2] that if two nite non-abelian groups G and H have the samenon-commuting graph, then jGj = jHj. This conjecture is refuted by an exampledue to I. M. Isaacs given in [4]. M. R. Darafsheh [4] has proved that the conjectureis valid whenever one the groups G or H are simple. Abdollahi et al. [1] showedthe validity of the conjecture whenever one the groups G or H has prime power

2010 Mathematics Subject Classication. Primary 20D60; Secondary 20F99.Key words and phrases. Non-commuting graph; Nilpotent group; Finite group.

28

order. In [3] it is proved that the conjecture holds if the non-commuting graphs ofthe groups are irregular. Note that in the example given by Isaacs, groups have thesame regular non-commuting graph.

3 Nilpotent Conjecture

It is not known if G is a nite non-abelian nilpotent group such that G = H forsome H, then H is also nilpotent. In [2] it is noted that by using a result of [4] thelatter is true if jGj = jHj.

4 Nilpotency class conjecture

It is not known if two nite non-abelian nilpotent groups with same non-commutinggraphs have the same nilpotency class. The least example of two nilpotent non-isomorphic groups with the same non-commuting graphs are dihedral group of order8 and the quaternion group of order 8, where these two latter groups have the samenilpotency class 2. By examples given in [4] one can see that for every prime p 3there are groups G1 and G2 of order p

5 such that G1= G2 such that the nilpotency

class of G1 is 2 and the nilpotency class of G2 is 3. Let us bring the details from [4].Let G be a non-abelian nite p-group possessing an abelian maximal subgroup M(necessarily normal of index p). Observe that, for x 2 M n Z(G), CG(x) = Mand if x 2 G nM , then CG(x) = Z(G)hxi. It follows that all proper centralizersof G are abelian and their orders are jGj or jGj=p or pjZ(G)j. Therefore the non-commuting graph of G is the complete multipartite graph KjGj=p;pjZ(G)j;:::;pjZ(G)j,where the number of parts of order pjZ(G)j is equal to jGjpjZ(G)j . Hence, if G1 and G2are two non-abelian nite p-groups possessing abelian maximal subgroups such thatjG1j = jG2j and jZ(G1)j = jZ(G2)j, then G1 = G2 .

Back to the non-abelian p-group G possessing an abelian maximal subgroup Mand suppose further that the subgroup M is elementary abelian, so that we canregard it as a vector space over Fp, the eld with p elements. The element x inducessome linear transformation on this space, and if t is the number of blocks of theJordan form of this transformation, then jZ(G)j = pt. The class of G, on the otherhand, is the maximal size of these blocks. Now each Jordan block has size at mostp, and so if p = 2, the partition into blocks is uniquely determined by their numberand the Fp-dimension of M . However, if p 3 and dim(M) = 4, it is easy toconstruct examples of linear transformations x and x0 of M with the same numberof blocks but having dierent maximal sizes. (For instance, when dim(M) = 4, thelinear transformation x could have two blocks of size 2 while x0 has a block of size 1and a block of size 3.) Then the semidirect products G1 = Mhxi and G2 = Mhx0i

29

have the same non-commuting graphs. In particular, there is a pair of such groupsof order p5 for any p 3.

5 Solvable conjecture

We do not know if G is a nite non-abelian solvable group such that G = H forsome H, then H is also solvable.

Acknowledgement

This research was in part supported by a grant from IPM (No. 92050219).

References

[1] A. Abdollahi, S. Akbari, H. Dorbidi and H. Shahverdi, Commutativity patternof nite non-abelian p-groups determine their orders, Comm. Algebra, 41 No. 2(2013) 451-461.

[2] A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group,J. Algebra, 298 (2006) 468-492.

[3] A. Abdollahi and H. Shahverdi, Non-commuting graphs of nilpotent groups, toappear in Comm. Algebra.

[4] J. Cossey, T. Hawkes, A. Mann, A criterion for a group to be nilpotent, Bull.London Math. Soc., 24 (1992) 327-332.

[5] M. R. Darafsheh, Groups with the same non-commuting graph, Discrete AppliedMath., 157 No. 4 (2009) 833-837.

[6] A. R. Moghaaddamfar, About noncommuting graphs, Siberian Math. J., 47 No.5 (2006) 911-914.

30

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

The relative nth nilpotency degree of two subgroupsof a nite group

Muhanizah Abdul Hamid1, Nor Muhainiah Mohd Ali2, Nor HanizaSarmin3 and Ahmad Erfanian4

1;2;3 Department of Mathematical Sciences, Faculty of Science, Universiti TeknologiMalaysia, 81310 UTM Johor Bahru, Johor, Malaysia

muhanizah.maths@gmail.com1, normuhainiah@utm.my2, nhs@utm.my3

4 Department of Mathematics, Faculty of Mathematical Sciences, Ferdowsi University ofMashhad, Mashhad, Iran

erfanian@um.ac.ir

Abstract

The commutativity degree of a group is the probability thattwo randomly chosen elements of G commute. The conceptof commutativity degree is then extended to the relativecommutativity degree of a group, which is dened as theprobability that two arbitrary elements one in H and an-other in G commute. Similarly, we can extend it to twoarbitrary elements one in H and another in K, where Hand K are two subgroups of G. In this research, the rel-ative commutativity degree concept is further extended tothe relative n-th nilpotency degree of two subgroups of agroup G which is dened as the probability that the com-mutator of two arbitrary elements h 2 H and k 2 K belongto Zn(G), where Zn(G) is the nth central series of G. Wegive some upper and lower bounds for the about probabilityand compute it for some known groups.

Speaker2010 Mathematics Subject Classication. Primary 16U80; Secondary 20F99, 20D15.Key words and phrases. Commutativity degree, relative commutativity degree, relative n-th

nilpotency degree of two subgroups.

31

1 Introduction

The theory of commutativity degree in group theory is one of the oldest areas ingroup theory and plays a major role in determining the abelianness of the group.It has been attracted by many researchers and it is studied in various directions.Many papers give explicit formulas of the commutativity degree of G denoted byP (G) for some particular nite groups G. The concept of commutativity degreecan be generalized and modied in many directions. For instance, commuting of anelement of a subgroup H with an element of G or even commuting of two elementson in subgroup H and another in subgroup K of G ; or by changing the role ofcommuting the rst element in subgroup H to the n-th power of such element. Onedierent way of generalization of commutativity degree is to replace the notion ofcommutativity by nilpotency class n. It is clear that if n = 0 then two concepts ofnilpotency class 0 and commutative are concide. So, we may associate parameter nto this kind of generalization of commutativity degree. First, let us remind that ifG is a nite group, then the commutativity degree of G, denoted by P (G), is theprobability that two randomly chosen element of G commute. The rst appearanceof this concept was in 1944 by Miller [5]. Then, the idea to compute P (G) forsymmetric groups has been introduced by Erdos and Turan [3] at the end of the 60s.

For any nite group G and its subgroup H, the relative commutativity degreeof G denoted by PG(H;G), is dened as the probability for an element of H and anelement of G commute. This concept was rst introduced by Erfanian et al: in [3].Similarly, if K is another subgroup of G then we may extend it as the probabilityfor an element of H commute to an element of K which is denoted by PG(H;K).This probability is called the relative commutativity degree of two subgroups H andK of a group G which can be written as

PG (H;K) =jf(h; k) 2 H Kjhk = khgj

jHj jKj :

In 2011, Erfanian et al. [4] dened the relative n-th commutativity degree de-noted by Pn(H;G) as the probability that the n-th power of a random element of Hcommutes with a random element of G. Now, we are going to give a generalizationof commutativity degree, relative commutativity degree of a subgroup and relativecommutativity degree of two subgroups in a dierent way as the following. For anygroup G and two subgroups H and K of G dene

Pnil(G) (n;H;K) =jf(h; k) 2 H Kj[h; k] 2 Zn1(G)gj

jHj jKj ;

32

which is called the relative n-th nilpotency degree of two subgroups H and K in G.If K = G then is denoted by Pnil(G)(n;H;G) and called the relative n-th nilpotencydegree of subgroup H in G and if H = K = G then is denoted by Pnil(G)(n;G) andcalled the n-th nilpotency degree of G. It is clear that if n = 1 then Pnil(G)(1; G) =P (G), Pnil(G)(1;H;G) = PG(H;G) and Pnil(G)(1;H;K) = PG(H;K). Morover,Pnil(G)(n;G) = 1 if and only if G is nilpotent of class n. Also, if H Zn(G) thenPnil(G)(n;H;G) = 1.

2 Main results

The following theorems give a lower and upper bound for the above probabilties.

Theorem 2.1. Let G be a nite group, H be a subgroup and p is the smallest primenumber dividing the order of G. Then for every n 1

jZn(G)jjGj +

p(jGj jZn(G)j)jGj2 Pnil(G)(n;G)

jHj+ jZn(G)j2 jHj :

We can also improve it for Pnil(G)(n;H;G) and Pnil(G)(n;H;K).

Theorem 2.2. If G is a nite group and H is a subgroup of G. Then for everyn 1

jZn(G) \HjjGj +

p(jHj jZn(G) \Hj)jGj2 Pnil(G)(n;H;G)

jHj+ jZn(G) \Hj2 jHj ;

where p is the smallest prime number dividing the order of G.

The following theorem gives a comparison of the probability of G and GN .

Theorem 2.3. Let G be a nite group, H and N be subgroups of G such thatN H. If N is normal then for every n 1

Pnil(G)(n;H;G) Pnil(GN)(n;

H

N;G

N)Pnil(N)(n;N):

Finally, we state some evaluations ofthe above probabilities for some small groupsand small values of n through the group theory package GAP.

33

Acknowledgement

The authors would like to thank Universiti Teknologi Malaysia (UTM) for the -nancial funding through the Research University Grant (RUG) Vote No. 04H13and UTM Mobility Program. The rst author would also like to thank Ministry ofEducation (MOE) Malaysia for her MyPhD Scholarship.

References

[1] D. S. Dummit and R. M. Forte, Abstract Algebra, Third Edition, USA. JohnWiley and Sons, Inc., 2004.

[2] P. Erdos, and P. Turan, On some problems of a statistical group theory, IV, ActaMath. Acad Sci. Hungaricae, 19 (1968), 413-435.

[3] A. Erfanian, B. Tolue and N. H. Sarmin, Some considerations on the n-th com-mutativity degrees of nite groups, Ars Combinatorial Journal. Press, 2011.

[4] A. Erfanian and B. Tolue, Relative non nill-n graphs of nite groups, ScienceAsia38 (2012), no 1, 201 - 206.

[5] A. Erfanian, R. Rezaei and P. Lescot, On the relative commutativity degree of asubgroup of a nite group, Communications in Algebra, 35 (2007), 4183-4197.

[6] G. Miller, A relative number of non-invariant operators in a group, Proc. Nat.Acad. Sci. USA, 30 (1944), no. 2, 25-28.

34

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

Generalize commutator on polygroups and hypergroups

Gholamhossien Aghabozorgi1, Morteza Jafarpour1 and Bijan Davvaz2

1 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

h.aghabozorgi@vru.ac.ir, m.j@vru.ac.ir

2 Department of Mathematics, Yazd University, Yazd, Iran

davvaz@yazduni.ac.ir

Abstract

The purpose of this paper is to provide a detailed structuredescription of derived subpolygroups of polygroups. We in-troduce the concept of perfect and solvable polygroups andwe give some results in this respect.

1 Introduction

Hyperstructure theory was born in 1934 at the 8th congress of Scandinavian Math-ematicions, where Marty introduced the hypergroup notion as a generalization ofgroups and after, he proved its utility in solving some problems of groups, algebraicfunctions and rational fractions. Surveys of the theory can be found in the booksof Corsini [3], Davvaz [4], Corsini and Leoreanu [4]. In the following we generalizeCommutator and dened derived subpolygroup. We recall here some basic notionsof hypergroup theory.

Let H be a non-empty set and P (H) be the set of all non-empty subsets ofH: Let be a hyperoperation (or join operation) on H, that is, is a function fromH H into P (H): If (a; b) 2 H H, its image under in P (H) is denoted bya b. The join operation is extended to subsets of H in a natural way, that is, fornon-empty subsets A;B of H, A B = [fab j a 2 A; b 2 Bg. The notation a A is

Speaker2010 Mathematics Subject Classication. 20N20.Key words and phrases. Hypergroup, Polygroup, derived subpolygroup, solvable polygroup,

Commutator.

35

used for fag A and A a for A fag. Generally, the singleton fag is identied withits member a. The structure (H; ) is called a semihypergroup if a (b c) = (a b) cfor all a; b; c 2 H, which means thatS

u2xyu z = S

v2yzx v;

and is called a hypergroup if it is a semihypergroup and aH = H a = H for all a 2 H.

A hypergroup P is called polygroup and is denoted by hP; ; e;1i if the followingconditions hold:

(1) P has a scalar identity e (i.e., e x = x e = x, for every x 2 P );(2) every element x of P has a unique inverse x1 in P ;

(3) x 2 y z implies y 2 x z1 and z 2 y1 x.A non-empty subset K of a polygroup hP; ; e;1i is a subpolygroup of P if x; y 2 Kimplies x y 2 K, and x 2 K implies x1 2 K. A subpolygroup N of a polygrouphP; ; e;1i is normal in P if x1 N x N , for all x 2 P:

2 Derived subhypergroups

In this section, we introduce and analyze a new denition for derivative of a hyper-group H.

Denition 2.1. Let H be a hypergroup. We dene

(1) [x; y]r = fh 2 H jx y \ y x h 6= ;g ;(2) [x; y]

l= fh 2 H jx y \ h y x 6= ;g ;

(3) [x; y] = [x; y]r [ [x; y]l :From now on we call [x; y]r , [x; y]l and [x; y] right commutator x and y, left

commutator x and y and commutator x and y, respectively. Also, we will denote[H;H]r , [H;H]l and [H;H] the set of all right commutators, left commutators andcommutators, respectively.

Proposition 2.2. IfH be a group, then [y; x]1r

= [x; y]r = [x1; y1]

l= [y1; x1]1

l;

for every x; y in H:

36

Example 2.3. Suppose that H = fe; a; bg. Consider the hypegroup (H; ), where is dened on H as follows:

e a be a; b e ea e a bb e a; b a; b

It is easy to see that fag = [a; a]r 6= [a; a]l = fa; bg = [a1; a1]l; where a1 isthe inverse of a in H:

Proposition 2.4. If H is a commutative hypergroup, then [x; y]r = [x; y]l = [x; y],for all (x; y) 2 H2:

Let X be a nonempty subset of a polygroup hP; ; e;1i. Let fAij i 2 Jg bethe family of all subpolygroups of P in which contain X. Then \i2JAi is called thesubpolygroup generated by X. This subpolygroup is denoted by < X > and wehave < X >= [fx"11 : : : x"kk j xi 2 X; k 2 N; "i 2 f1; 1gg. If X = fx1; x2; : : : ; xng,then the subpolygroup < X > is denoted < x1; x2; : : : ; xn >. In a special case< [P; P ]r >, < [P; P ]l > and < [P; P ] > are shown by P

0r, P 0

land P 0, respectively.

Proposition 2.5. Let hP; ; e;1i be a polygroup (x; y) 2 P 2: Then,(1) [x; y]r = [x

1; y1]l;

(2) P 0 = P 0r= P 0

l;

(3) x 2 P 0 ) x1 2 P 0.Corollary 2.6. If hP; ; e;1i is a polygroup, then P 0 is a subpolygroup of P:

From now on we call P 0 the derived subpolygroup of P:

Proposition 2.7. Let hP; ; e;1i be a polygroup. Then, P 0 = feg if and only if Pbe an abelian group.

Denition 2.8. A polygroup P is called perfect if and only if P 0 = P .

Denition 2.9. A polygroup P is called solvable if and only if P (n) = !P , for somen 2 N, where P (1) = P 0, P (n+1) = (P (n))0 and !P is heart of polygroup P .Proposition 2.10. Every non-trivial perfect group is not solvable.

In the following, we show that the above proposition is not true for the class ofpolygroups.

37

Example 2.11. Suppose that P = fe; a; b; cg. Consider the commutative polygrouphP; ; e;1i, where is dened on P as follows:

e a b ce e a b ca a P a; b; c a; b; cb b a; b; c P a; b; cc c a; b; c a; b; c P

We can easily see that P is a perfect and solvable polygroup. Notice that P 0 = P =!P .

Example 2.12. Suppose that P = fe; a; b; cg. Consider the non-commutative poly-group hP; ; e;1i, where is dened on P as follows:

e a b ce e a b ca a a P cb b e; a; b b b; cc c a; c c P

In this case, we can see that P 0 = P = !P .

Example 2.13. (Double coset algebra) Suppose that H is a subgroup of a groupG. Dene a system

G==H = hfHxHjx 2 Gg; ;H;Ii;where (HxH)I = Hx1H and (HxH) (HyH) = fHxhyHjh 2 Hg. The algebraof double cosets G==H is a polygroup.

Theorem 2.14. Let (G; ) be a group and H be a subgroup of G. We set HG0H =fHgHjg 2 G0g: Then,(1) HG0H (G==H)0;(2) If G0 H = G then (G==H) is a perfect polygroup;(3) If HG0H = (G==H) then G0 H = G.

References

[1] H. Aghabozorgi, B. Davvaz and M. Jafarpour, Solvable polygroups and derivedsubpolygroups, Comm. Algebra, 41(8)(2013) 3098-3107.

38

[2] H. Aghabozorgi, B. Davvaz and M. Jafarpour, Nilpotent groups derived fromhypergroups. J. Algebra 382 (2013) 177-184.

[3] P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, Tricesimo, 1993.

[4] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Aca-demical Publications, Dordrecht, 2003.

[5] B. Davvaz, Polygroup Theory and Related Systems, World Scientic, 2013.

39

The Extended Abstract of

The 6th International Group Theory Conference

12{13 March 2014, Golestan University, Gorgan, Iran.

Some solved and unsolved problems in loop theory

Karim Ahmadidelir

Department of Mathematics, College of Basic Sciences, Tabriz Branch, Islamic AzadUniversity, Tabriz, Iran

kdelir@gmail.com, k ahmadi@iaut.ac.ir

Abstract

In this talk, we consider some solved and unsolved problemsin the theory of loops and quasigroups and disscuss aboutrecent progresses and advances or improvements of them.Some of them are long-standing and well-known problemsthat newly have been solved and developed the theory andsome of them are the existing theorems in group theory thathave been generalized to some special kinds and classes ofloops. On the other hand, there have been achievements andimprovements in some of unsolved problems in recent yearsthat opened new horizons in the theory.

1 Introduction

A setQ with one binary operation is a quasigroup if the equation xy = z has a uniquesolution in Q whenever two of the three elements x; y; z 2 Q are specied. Loop isa quasigroup with a neutral element 1 satisfying 1x = x1 = x for every x. Moufangloops are loops in which any of the (equivalent) Moufang identities ((xy)x)z =x(y(xz)); x(y(zy)) = ((xy)z)y; (xy)(zx) = x((yz)x); (xy)(zx) = (x(yz))x holds.

Moufang loops are certainly the most studied loops. They arise naturally in al-gebra (as the multiplicative loop of octonions), and in projective geometry (Moufangplanes), for example. Although Moufang loops are generally nonassociative, theyretain many properties of groups that we know and love. For instance: (i) every x

2010 Mathematics Subject Classication. 20N05.Key words and phrases. Theory of loops and quasigroups, Moufang loops, Bol loops, Bruck

loops, A-loops.

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is accompanied by its two-sided inverse x1 such that xx1 = x1x = 1, (ii) anytwo elements generate a subgroup (this property is called diassociativity), (iii) innite Moufang loops, the order of an element divides the order of the loop, and ithas been shown recently in [4] that the order of a subloop divides the order of theloop, (iv) every nite Moufang loop of odd order is solvable.

On the other hand, many essential tools of group theory are not available forMoufang loops. The lack of associativity makes presentations very awkward andhard to calculate, and permutation representations in the usual sense impossible.

The other most studied classes of loops are: Bol loops, Bruck loops, Osborn loops,RS-loops, A-loops. In this talk, we consider some solved and unsolved problemsabout these special classes of loops and disscuss about recent progresses and advancesor improvements of them.

2 Basic concepts

Let Q be a loop with neutral element 1. We dene:

Left multiplcation operator by: Lx : Q!! Q; y 7! x y; Right multiplcation operator by: Rx : Q!! Q; y 7! y x,(Lx and Rx are bijections of Q)

Commutator of x and y by: = [x; y] : xy = (yx) [x; y]; Associator of x; y and z by: = [x; y; z] : (xy)z = x(yz) [x; y; z]; Commutant of a subset S of Q by: fx 2 Q j xs = sx; 8s 2 Sg; Center of Q by: Z(Q) = fx 2 Q j [x; y] = [x; y; z] = [y; x; z] = 1g; Multiplication group of Q by: Mlt(Q) = hLx; Rx j x 2 Qi; Inner mapping group of Q by: Inn(Q) = ff 2Mlt(Q) j f(1) = 1g; A subloop S Q is normal if f(S) = S for every f 2 Inn(Q); The nucleus Q by: fx j x(yz) = (xy)z, y(xz) = (yx)z, y(zx) = (yz)x; 8y; z 2Q;

A loop Q is solvable if there is a series 1 = Q0EQ1E EQm = Q such thatQi+1Qi

is an abelian group for every i;

p-loops by: Loops of order pk, p a prime;

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A nite Moufang loop Q is a p-loop if and only if every element of Q has orderthat is a power of p;

A loop Q is (centrally) nilpotent if the sequence Q; QZ(Q) ;Q

Z(Q)

Z( QZ(Q)

); : : : eventually

yields the trivial loop;

Let Q be a centrally nilpotent p-loop. The Frattini subloop (Q) of Q is theintersection of all maximal subloops of Q;

An isotopism of loops Q1, Q2 is a triple (; ; ) of bijections Q1 ! Q2 suchthat (x)(y) = (xy) holds for every x; y 2 Q1;

G-loop: all isotopes of a loop Q are isomorphic to Q; Right Bol loops: the loops are given by the right Bol identity x((yz)y) =((xy)z)y (Left Bol loops are dened analogously);

Bruck loop orKloop: a Bol loop satisfying the automorphic inverse property,(ab)1 = a1b1 for all a; b in Q.

We refer the reader to [1] and [6] for a systematic introduction to the theory ofloops.

3 Some open problems about Moufang loops

Problem: Let p and q be distinct odd primes. If q is not congruent to 1 modulo p,are all Moufang loops of order p2q3 groups? What about pq4? (Proposed by AndrewRajah at Loops '99, Prague 1999)Comments: The former has been solved by Rajah and Chee (2011) where theyshowed that for distinct odd primes p1 < < pm < q < r1 < < rn, all Moufangloops of order p21 p2mq3r21 r2n are groups if and only if q is not congruent to 1modulo pi for each i.Phillips' problem: (Odd order Moufang loop with trivial nucleus) Is there a Mo-ufang loop of odd order with trivial nucleus? (Proposed by Andrew Rajah at Loops'03, Prague 2003)Problem: (Presentations for nite simple Moufang loops) Find presentations forall nonassociative nite simple Moufang loops in the variety of Moufang loops. (Pro-posed by P. Vojtechovsky at Loops '03, Prague 2003)Comments: It is shown in (Vojtechovsky, 2003) that every nonassociative nitesimple Moufang loop is generated by 3 elements, with explicit formulas for thegenerators.

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Conjecture: (The restricted Burnside problem for Moufang loops) Let M be anite Moufang loop of exponent n with m generators. Then there exists a functionf(n;m) such that jM j < f(n;m): (Proposed by Alexander Grishkov at Loops '11,Tret 2011)Comments: In the case when n is a prime dierent from 3 the conjecture wasproved by Grishkov. If p = 3 and M is commutative, it was proved by Bruck. Thegeneral case for p = 3 was proved by G. Nagy. The case n = pm holds by theGrishkov-Zelmanov Theorem.Conjecture: (The Sanov and M. Hall theorems for Moufang loops) Let L be anitely generated Moufang loop of exponent 4 or 6. Then L is nite. (Proposed byAlexander Grishkov at Loops '11, Tret 2011)Conjecture: Let L be a nite Moufang loop and (L) Frattini subloop of L. Then(L) is a normal nilpotent subloop of L. (Proposed by Alexander Grishkov at Loops'11, Trest 2011)Conjecture: (Torsion in free Moufang loops) Let MFn be the free Moufang loopwith n generators. MF3 is torsion free but MFn with n 4 is not. (Proposed byAlexander Grishkov at Loops '03, Prague 2003)Problem: (Minimal presentations for loops M(G; 2)) Find a minimal presentationfor the Moufang loop M(G; 2) with respect to a presentation for G. (Proposed by P.Vojtechovsky at Loops '03, Prague 2003)Comments: Chein showed in (Chein, 1974) thatM(G; 2) is a Moufang loop that isnonassociative if and only if G is nonabelian. Vojtechovsky (2003) found a minimalpresentation for M(G; 2) when G is a 2generated group.

4 Some solved and Open problems about Bol loops

Remark 4.1. We have the following implications:

Right Bol loops(= Right and left Bol loops = Moufang loops(= groups.

Problem: (Existence of a nite simple Bol loop) Is there a nite simple Bol loopthat is not Moufang? (Proposed at: Loops '99, Prague 1999, Solved by: Gbor P.Nagy, 2007)Solution: A simple Bol loop that is not Moufang will be called proper. There areseveral families of proper simple Bol loops. A smallest proper simple Bol loop is oforder 24 [Nagy 2008]. There is also a proper simple Bol loop of exponent 2 [Nagy2009], and a proper simple Bol loop of odd order [Nagy 2008].Comments: The above constructions solved two additional open problems:Problem: Is there a nite simple Bruck loop that is not Moufang? Yes, since anyproper simple Bol loop of exponent 2 is Bruck.

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Problem: Is every Bol loop of odd order solvable? No, as witnessed by any propersimple Bol loop of odd order.Problem: (Left Bol loop with trivial right nucleus) Is there a nite non-Moufangleft Bol loop with trivial right nucleus? (Proposed at Milehigh conference on quasi-groups, loops, and nonassociative systems, Denver 2005, Solved by: Gbor P. Nagy,2007)Solution: There is a nite simple left Bol loop of exponent 2 of order 96 withtrivial right nucleus. Also, using an exact factorization of the Mathieu group M24,it is possible to construct a non-Moufang simple Bol loop which is a Gloop.Problem: (Nilpotency degree of the left multiplication group of a left Bol loop)For a left Bol loop Q, nd some relation between the nilpotency degree of the leftmultiplication group of Q and the structure of Q. (Proposed at Milehigh conferenceon quasigroups, loops, and nonassociative systems, Denver 2005)

A loop is universally exible if every one of its loop isotopes is exible, that is,satises (xy)x = x(yx). A loop is middle Bol if every one of its loop isotopes hasthe antiautomorphic inverse property, that is, satises (xy)1 = y1x1.Problem: (Universally exible loop that is not middle Bol) Is there a nite, uni-versally exible loop that is not middle Bol? (Proposed by Michael Kinyon at Loops'03, Prague 2003)Problem: (Finite simple Bol loop with nontrivial conjugacy classes) Is there anite simple nonassociative Bol loop with nontrivial conjugacy classes? (Proposedby Kenneth W. Johnson and Jonathan D. H. Smith at the 2nd Mile High Conferenceon Nonassociative Mathematics, Denver 2009)

5 Other problems

Much more problems about special kinds of loops have been solved or modiedrecently and published in various journals those will be presented and discussed inthis talk, such as: a problem about Enumerating Nilpotent Loops up to Isotopy, by L.Clavier (2012); a problem about dierences and similarities between Bruck loops andnite groups: Do nite Bruck loops behave like groups?, by B. Baumeister (2012);a problem about On RS-loops by P. Sclifos (2011); a problem about Necessaryconditions for the existence of the nite Osborn loop with trivial nucleus by T.G.Jaiyeola, J.O. Adeniran and A.R.T. Solari, (2011); a problem about Universality ofOsborn loops by M. Kinyon (2005); a problem about Associativi