QUASIIDENTITIES AND DIRECT WREATHS OF GROUPS

A. I. Budkin UDC 519.48

In this article we study quasivarieties of groups that are closed with respect to direct

wreaths. Examples of such quasivarieties are quasivarieties of ordered groups, quasivari-

eties generated by all finite groups, all finite p-groups, all soluble groups, and certain

others [3].

The question naturally arises as to whether any quasivariety of groups, closed with re-

spect to direct wreaths, can be defined by a system of quasiidentities, each of which is

stable with respect to direct wreaths. Let ~U/Z~ be the least variety closed with respect

to direct wreaths and containing the group ~ . In this article we answer the above question

for the quasivariety ~r~ ~ if ~ is a finite group, a torslon-free cyclic group, a torsion-

free nilpotent group, or a group with one defining relation. We calculate the cardinality

of the set of quasivarieties of groups, closed with respect to direct wreaths, each of which

is not defined by quasildentitles which are stable with respect to direct wreaths.

i. Preliminary Information

We recall the definition of the direct wreath of the two groups A and B. Take the

direct power A of the group A, consisting of all functions {:~-'~A with finite carrier. For

each ~6 ~ we define a mapping ~:f-~f~by the rule ~(~)= f(~-i) for all ~6~ . The

mapping ~ is an automorphism of the group ~ , and the set of all such automorphisms is a

group isomorphic to ~. We extend the group ~ using this group of automorphisms, and the

result is called the direct wreath of the groups ~ and ~ and is written A Z~. The group

is called the basis subgroup of the wreath.

We say that the class ~ of groups is closed with respect to direct wreaths if for any

groups A and ~ in ~ their direct wreath A~ also belongs to the class ~. The quasi-

identity ~ is called stable with respect to direct products if for any groups ~ and ~ , if

qO is true in the groups ~ and ~ then it is also true in the group ~ •

We denote the quasivarlety generated by the class ~ of the groups by 9~. The least

quasivariety closed with respect to direct wreaths and containing the class ~ of groups is

denoted by ~ . If A is a group, then instead of ~{~}, ~/~A~ we shall write g~ and

~U/Z ~ , respectively.

Let ~ be a class of groups. We say that the group ~ is ~-approximable if for any

nonunit element ~ there exists a normal subgroup N~& such that ~ and ~/~ is

isomorphically embeddable in some group in ~.

If G is a group, and A is a subset of ~ , we denote by gr CA) the subgroup generated by

the set ~ in ~. As usual, Z is the infinite cyclic group and ~ is the cyclic group

Translated from Algebra i Logika, Vol. 23, No. 4, pp. 367-382, July-August, 1984. inal article submitted March 15, 1984.

Or ig -

0002-5232/84/2304-0253508.50 © 1985 Plenum Publishing Corporation 253

of order r&. If ~ , ~ are elements of the group ~ , then we set

The commutant of the group ~ is denoted by ~I.

The basic properties of quasivarieties can be found in [4, 5].

We shall need the following well-known theorem of Dirichlet (see, for example, [7]):

any arithmetical progression

where the numbers a and ~ are coprime, contains an infinite set of prime numbers.

We shall need the following test for whether a finitely defined group G belongs to the

quasivariety generated by the class ~ of groups; this is a special case of Theorem 3 of [4]:

The finitely defined group ~ belongs to the quasivariety generated by the class ~ of groups

if and only if ~ is ~-approximable.

We shall use the following variant of Lemma 3 of [2].

LEMMA I. If the quasiidentity ~ is true in the group Z, then the set /={pl ~ is

false in ~p, where p is prime ~is finite.

To study the quasivariety ~4~'Z~, we shall use the following construction (Corollary 5

of [3]).

LEMMA 2. Let ~ be an arbitrary class of groups

Then 9 ~ ~ = 9 ~" All the quasiidentities we consider will be nontrivial, i.e., are not equivalent to the

identity (V.~) (.~ = 1). We denote by ~ the quasivariety of groups defined by all nontrivlal quasiidentities

which are stable with respect to direct wreaths.

2. Quasivariety

Let ~ be a nontrivial quasiidentity stable with respect to direct wreaths. It is

easily seen that ~ is true in the group Z. If ~ is a group approximable by finite p-

groups for an infinite set of prime p, then it follows from L~-,ma 1 that ~ is true in ~,

and therefore &E~. In particular, the quaslvariety~ contains all torslon-free polycyclic

groups, and all torslon-free nilpotent groups. On the other hand, the quasiidentlty

is stable with respect to direct wreaths. Therefore, the quasivariety ~does not contain

any groups with torsion.

Our first aim is to construct, for each natural number ~2) the group ~a ' and to

show that G. ~ ~.

254

Take two groups ~ and C , each of which is isomorphic to the additive group of ~-ary

fractions. Let

a = <..., 4 . - g , ' ,-- o, ),

C=g, (. . . , c_, , co, c, , l c ; + , = : . . . ; , i = o , + - / , + - ~ , . . . ) ,

It is easily seen that the mapping

, o z ~ o a , i = o , ± ~ ±~,...,

can be e x t e n d e d to an au tomurph i sm a o f t he g r o u p ~ O , and t h e mapping

can be extended to an automorphism ~ of the group ~ . We denote the image of the element

~ under the mapping ~ by {~)~. We denote by ~ the extension of the group ~ by means

of the group of automorphisms ~ = gr (a,~),

LEMMA 3. Let ~={~1~.,.,~I~ #~ for ~#j~ be a finite set of prime numbers, and

let ~M be the set of all finite soluble groups such that none of the numbers in ~ is a

prime divisor of the order of any group in M . Then the group ~ is ~N -approximable.

Proof. We first write the elements of the group ~ in some sort of "canonical" form.

By direct verification we see that

= K i_4,..Ci_zx , x~/, (3)

It follows from (1)-(3) that

~ ~a, dJ = 6i ci_ , , ci-~ • ~ It follows from (4) that

[ a , d ] = 0iC;_4¢ , K - is an]/" integer, ( 5 )

(~) ~-'co,~; e~= 4 _, ,~. ,e ~i-gX-4' K, ~ are any integeR, (6)

r6~ ) d -~ ca,='~ed ~=~ , ,_.z ,e Oigi_gx ~ , ~,-e are any integet~, (7)

It follows from (6) and (7) that the following relatiob is true in the group A :

- x ~ ~ - x a ~,a3a =d Ea, a~d , K is~yin~go,. ~8)

We introduce the notation z , , = d - " [ a , d ] d '~ .

any g r o u p ,

{ Vx,//, z)CFa-/,,,z] = i-' [~, z]i[i,z]), A' then it follows from (8) that the commutant of the group A is equal to

Since the following identity is true in

255

A;= sr (z~ l ~=o , -+ 4 -+;, . . • ) . (9)

It follows from (7) that

Thus,

Since (5)

rive group of n-ary fractions. From (i),

~2 =,+ : z,~_, • (lO)

A' A' implies that is a torsion-free group, we note that is isomorphic to the addi-

(2), and (7) we have the equation

/+ez+~++... + ~ ~z) cd

Hence we see that the following relation holds in the group A :

~a-/~/-aZ S %-z = t '

There fo re , in view of (9) , any element of the group #a can be written in the form

It follows from (Ii) that we may assume that IZTZI<~z--I , and from (i0) we may assume that

(ii)

~=0 or ~ is not divisible by /z z .

Take an element ~E~ , ~{.

- ~ - , , + - ~ ~.~,c,. s •

be the canonical notation of the element ~ ~/7Z l</Zz-/, {--~

Our aim is to construct a group ~e ~M and a homomorphism

Take a prime number ~ such that

We call this notation of elements canonical.

Let

or ~ is not divisible by ~z ).

~:G.-+# such that (~)~÷i.

(12)

g >mo.z{~,,~a,...,g#, Is, I, I%1,1~1,1#1,1{ I } , (13)

> (¢+~%~%... +~z,~+2f?i+zI~,+) zlt~ + l~-~'J l{Is zltj+t'~s++ . (14)

The existence of ~ follows from Dirichlet's theorem.

Let ~ = gr {~t), ~= gr (6") be cyclic groups of order ~, and let ~=~x~ . 4z

It is easily seen that the mapping ~-~ b ~ , b~--~b ~ can be extended to an automorph-

ism ~ of the group US. Analogously, the mapping ~ --~ 4z ,4/'---~" can be extended to an auto-

morphlsm ~ of the group ~r.

Let ~= gr (o~,~} , and let ~ be an extension of the group ~r by means of the group of

automorphisms ~ .

Since for any ~ ~ ~ the equation ~= ~ has a unique solution, then we can uniquely

define elements ~d"~ E ~ such that

256

It is easily seen that

o"o = = 6 " ,

~ - 0,+/. +2 '

~ = 0 , ± t,~,~, . . . .

Hence it follows that if (~)~((2,d)=~' , then (~)~,~)=~" . In other words, if the

relation %(~,~)=/ is satisfied in the group A, then the relation Z~,#)=/ is satisfied in

S. Thus, the mapping a ~ , ~--'~ can be extended to a homomorphism of the group

into the group S . It is easily seen that this homomorphism can be extended to a homomor-

phism ~: ~"~, where

We show that ( } ) ~ : / . Suppose that (~)~=/ , i.e.,

i sln:e ~i~/, ~#/, 9>Is, l, ~ >lS~l ,weseethat ~/=S£=O. Therefore, we have

=/.

Case 1. /TI ~ t . By analogy with (I), (2), and (7), we have

£(/-n a) (%)~ /~

/z 2nZ+ 2~ Uz.,+2, ~ = u ° fo, Zrn+2~ ~0,

/Z 12m+2,< I 1~o=(,Z~4.2~ for 2nz4-2K < 0 ,

Thus ~2m+2K =dO " But

( is)

Hence it follows that /~lbn+2Kl 12m+ZKI = i ( n 2 0 ~ ) . By property (14) we see that f& = / , and

thus /7~+K = 0 . Therefore, it follows from (15) that

Zm- 2 .1+nz+~*+, . + /z ~ ( t - ~ z z )

% ' ~-ef-, = / ' (16)

r . e . ,

, we have

257

Hence we see that

By property (14) we obtain

But this equation is false,

Therefore, 2~-~0 .

that

Thus,

/+n%,,z + + ... + ~ " - ~ + /f~f-~ { C/_~)--o(m.adz ).

/~,~%~+,.. ~ z~-z+ f[zf-~U_~ b = O.

since 1 is not divisible by ~ .

Suppose that 2f-4 <0 In this case it follows from (16)

1~,{+,41, z ÷ . +/Z 2, '-z = 4 w,,or~ Z~,= ~f~'4-,',')+,'Z, (,,+~ +~ ,-.. ).

Once again it follows from (16) that

... + , ~ ' - ~ ) - = o (~dg).

iz~+41 ~.-z ) l(1_,z) + ~ (~+~'+M+... + ~ = 0 .

Hence we see that ~ is divisible by /~2. By the property of the canonical notation, this

means that #=0 , i.e., the last equation can be rewritten thus:

~ ~ 2 {+4 ! .. ~ 2,, ,-2 ~ (/+nz+ . + ) = I.

But this is impossible. Thus, we have shown that Zf-~ =O , i.e., /=-Z .

Eq. (16) can be rewritten thus:

Hence, as before, we obtain /+/Z2+,., +/~Z'~-2+~[t'-IZz]=O. Thus,

/~ nz+... +/Z z~-a (/z~_Z)+(n~_/)+...+(~a~z_l) = +

= /Tz_l nz-/ tzJ-/ •

Since ~ is an integer, I/TIl <~Z._/ , this equation is impossible.

that ~0.

Case 2. 17L~-/.

By analogy with the above, we obtain

- , . ,

~. . . . ~,,.,, ~,:,,+~_.,f_.~ = a o . As before, we have K+~=0 . Thus, from this relation we have

~r_.i -I -i ~r (i-~91 2 ~-~ "'" ~-~ -z~-~ = / "

In this case,

Therefore, we have shown

(17)

If 2f-~<2/~ then from (17) we obtain

258

As before, by the choice of ~ we obtain

(~-~)~- (~+z+ azt+. +~+4+z~)= O.

This means that ? is divisible by fl.. z and, therefore, ~--0 •

equation is impossible.

If ~-~ ~ then from (17) we obtain ~)'~ = { where

+ l}+E'~-¢-~"4t~%

Thus, we see that the above

~, = _ ( ~ - ~ - ~ , + ~ - ~ - " +...

Once again, as before, we obtain

-(rJ-~'* + rf *-~ +... +I) + rf,~P~-~'*~ t~-~)=O.

This equation is impossible, in view of the fact that the left-hand side is not divisible by

.

Therefore, -2~-#=2n~ . In this case, we have

E~' = t "h°'° 6 = - (t+,~ ~ + ' ~ + . . - + 4 " - ~ ) + t ~O~.

Hence, by the choice of ~ we obtain

(~-~)~ - (4+ a ~ + ~ +. Thus,

.. + E '~ ' ' -~) = O .

l--tZ ~ fl I--t ~ - t '

This equation is impossible, since ~ is an integer and In~l <a ' - I .

Therefore, we have shown that cases i and 2 are impossible. Thus,

Eq. (15) can be rewritten as follows:

= u ~ ~,p-,~+.,~,~ = u o .

and

Z( /-- /~ z~ -= O . Thus, ~'--0.

Hence it follows that ~=0

By the choice of ~ we have

Thus, (9)~i .

We now show that

Since

HE ~I~ " We note that

Therefore, the order of the element ~ divides

/~= 0 • In this case,

This means that 9 = I .

(i+az+... +~-3)(az-i} = ~-~_

we have /+l~Z+. • +rb ~-3" . ---o ( , ~ ) . 8_!. 2

Contradiction.

Hence it follows that (~)~'2- = ~ .

Analogously, the order of the element

259

divides ~-I. Since 2

(U)

then the order of [=L,~] is equal to ~. But SI=(AI)~ and A! is a locally cyclic group and,

therefore, the order of S t is equal to ~. It follows from the above that the order of the group /I is a divisor of the number ~'{~)'. But

and therefore ~/ is not divisible by a number in M. This means that H~p4. The

lemma is proved.

Proof. Let ~ be a nontrivial quasiidentity which is stable with respect to direct

wreaths. It is easily seen that 9~ is true in the infinite cyclic group Z . Let ~=~ I

is false in ~p , where /~ is prime~. By Lemma i, / is a finite set, and by eemma 3 ~ is

~l-approximable. In view of the stability of q~ with respect to direct wreaths, since the

truth of 9 6 in ~p~/ implies the truth of ~ in any group in ~/, we see that 9~ is

true in ~/z" Thus, ~ e~ . The lemma is proved.

The following remark allows us to construct new groups belonging to an arbitrary quasi-

variety, closed with respect to direct wreaths.

Proposition. Let the quaslvariety ~ of groups be closed with respect to direct

wreaths. If ~ ~ , then their free product A ~ ~Z.

Proof. Let ~ be the cartesian subgroup of the group ~#~, ~ m ~ , ~-~, and let

be the free abelian group of rank 2. It was proved in [8] that there exists an isomorphic

embedding of the group ~m~/~r into the group Ha~({~#~)/~) . Hence it follows that for

any integer ~{&>~O)(~*~)/C~)~ ~, where ~a~, ~{~,i)__ (~0 p But the cartesian subgroup

is a free group and, therefore, ~ ~(i) __ (f) . Thus, A ~ is approximated by groups I=0

(~a.~)/O ~) , ~=0,/,2, . . . . Thus, ~a..~e ~'~. The proposition is proved. ~z

COROLLARY 1. Let ~ = <a, ~ ; a " l a = >, it Then .

In fact, ~n is a subgroup of the group ~n •

3. Quasivariety ~r'6

We note the following obvious fact: if ~ , where ~ is a nonunlt quasivariety of

groups, then ~ cannot be defined by a system of quaslidentities, stable with respect to

direct wreaths.

THEOREM I. Let ~ be a finite nonunit group; then ~r~ ~ cannot be defined by a system

of quasildentities stable with respect to direct wreaths.

Proof. Let the order of ~be equal to /Z. Consider the group ~=<O,~; a-l~ ~-

~aa . By Corollary i, ~ . To prove the theorem, it is sufficient to show that

On the contrary, let ~Iz ~ g~/~ ~ , ~! = {~g ~,

260

By Lemma 2, ~ e ~ . By the test for the group ~ to belong to the quasivariety ~, we

see that the group ~ is ~-approximable. Therefore, there exists a homomorphism ~:~ --~

~£~ such that ( ~ / . Since the elements (~)~and (~#~ are conjugate, then their

orders coincide. But, on the other hand, the numbers n and IC~)~I , where I~)~l is the order

of the element (~, are not coprime and, therefore, [(~I<I[~. Contradiction. Thus,

~ ~ ~ ~ ~ The theorem is proved.

COROLLARY 2. The quasivariety ~ , generated by all finite ~ -groups (where ~ is a

fixed prime number) cannot be defined by a system of quasiidentities, stable with respect to

direct wreaths.

Proof. This follows from the fact that ~ = ~ r ~ ,

COROLLARY 3. Let ~ (~=~.,.,~) be a finite nonunit group. Then ~(~#~...-~)

cannot be defined by a system of quasiidentities stable with respect to direct wreaths.

Proof. By the Proposition we see that ~g(~1#~,..#~) = gG/%(~x~2x...X~) . By

Theorem i, ~r~(~×6×,..x~ ~) cannot be defined by a system of quasiidentities stable with

respect to direct wreaths.

THEOREM 2. Let ~r~ be a class of torsion-free groups, in each of which the following

quasiidentity is true:

e= = f

where /Z is a fixed natural number, /Z~>2. Then ~0~ ~ cannot be defined by a system of

quasiidentities stable with respect to direct wreaths.

Proof. We show that ~ is true in any group in 9U/~ ~ • In view of Lemma 2, it is

sufficient to show that if ~ and ~ are torsion-free groups and ~ is true in ~ and in ~,

then ~ is also true in ~ . Let the left-hand side of ~ be true in ~ for the

substitution ~-~, ~ --+ b~ , where ~,fE~, ~,~G~. Then we have

Since ~--~(~)/i , ~ is true in C~?J~)/i . Hence, in view of (18), we have

i.e., F=/. Thus, in the group ~ we have the relation

Suppose that ~ ~l . Since ~ is a torsion-free group, there exists an element

that ~(LE)~4, ~(~£,~-/) = / . Hence we obtain

Thus ~zI~)=/. But ~ is a torsion-free group and, therefore, ~(~)=~ Contradiction.

Hence, ~= ~ We have shown that ~ is true in ~ ?,~ , and thus ~ is true in any group

in ~ U/Z ~,

In view of Corollary i, ~=gr(G,~|a-/~G=~z)~. But ~ is false in ~n and,

;E6~ such

261

therefore, ~a¢~afZ~ . Thus, ~ ~z6rZ~ , and, therefore, ~#/'~ cannot be defined by

a system of quasiidentities stable w~th respect to direct wreaths. The theorem is proved.

COROLLARY 4. Let ~be a torsion-free group, any 2-generated subgroup of which is poly-

cyclic. Then ~ cannot be defined by a system of quasiidentities stable with respect

to direct wreaths.

In particular, as ~ we can take a torsion-free nilpotent group.

THEOREM 3. Let ~ be a nonunit group with one defining relation. Then ~Z ~ cannot be

defined by a system of quasiidentities stable with respect to direct wreaths.

Proof. In view of Theorem i, we may assume that ~is an infinite group.

Case i. ~ is a torsion-free group.

Consider the quasiidentlty

I t was shown in [1] ( the p roo f of Theorem 2) t h a t c~ i s t r u e in any group wi th one d e f i n i n g

relation. We now show that ~ is true in ~aTZ~. In view of Le-~a 2, it is sufficient to

show that if m and ~ are torsion-free groups and ~ is true in A and ~, then 9~ is true

in m~. This was proved in [3]. Thus, q~ is true in ~ •

We now take a 2-degree torsion-free nilpotent group ~, in which 95 is false. ~¢

~G/~ , but, as we remarked, ~ . Thus, ~ ~U/Z~ and, therefore, ~U/~ cannot be

defined by a system of quasiidentities stable with respect to direct wreaths.

Case 2. ~ is a group with torsion.

It was shown in [9] that the orders of elements of finite order in ~are all bounded.

Take a natural number ~ (n~ ~ such that ~ is divisible by the order of any element of fin-

ite order in ~. Consider the quasiidentity

We show that ~ is true in ~.

Let the left-hand side of the quasildentity q~ be true in 0 for the substitution

~-~Z, ~-w~, Z-~C, ~-~, C#/ Since by [i0] the centralizer of any element in the group

is a cyclic group, then we obtain

C /~/-'z~ /~/< = , O P = ~ .

Hence C = / , i . e . , ~K i s t r u e in ~ . We now show that c/~1¢ i s t r u e in ~LI~ 0 . Let a l l t he ~ (t~=~3,..o) be true in the

groups A and ~ , where the orders of all the elements of finite order in A and ~ divide the

number /~ra. In view of Lemma 2, it is sufficient to show that ~K is true in A ~ .

Let the left-hand side of the quaslidentity ~ be true in m~ for the substitution

- o f , * d / , ,

262

where G , < C , ~ 6 ~ i : i ~ i ~ , f # ~ f . Since ~ is true in A ~ / 7 , then we have C = / . Sup-

pose that ~I. The equation [~,~pl]={ implies that

f://'/f, : ( 1 9 )

zf d is an element of infinite order, then there e~ists a such that ,d:,~)/=I ,kt;zd -/) =/. It then follows from (19) that ~(~) = /. Therefore, d is an element of finite order, and,

in particular, d #~-- / . We prove analogously that a am= / •

We introduce the notation

i, i,</, o-= cd /= F, . . . ,

= : " ' , •. : f .

It follows easily from Eq. (19) that ~-~-~, and, therefore,

~--'k~-= ~" (2o) Analogously to Eq. (20), from the equation [~,~:~ ffi / we have the equation

~%~= k. (21)

The equation (d~)-%d~ = [:~)a~ implies that ~'~d = :~ . We note that it follows

from the equation [~,~3 / that : is an element of finite order, i.e., b e= {. Thus, ~=/. From the equation ((~'pf)'/~d: =~ ~ we now see easily that

where S ~l+~x+~2~ +.., + [6f ~-0~ ,

Thus, it follows from Eqs. (20), (21), (22), and (23) that the left-hand side of the

quasiidentlty ~K~ is true in the group ~ for the substitution

z---~, ~--~, ~--:, z-'/~ s.

Since ~ is true in the group A , we thus obtain ~&=/. Thus, ~ is an element of fin-

ite order. By the choice of the number ~, we have ~=/. Since the numbers S and ~ are

coprlme, we have ~={ . Thus, the quasiidentity ~ is true in ~D~.

We note that ~Z is false in the group &~ for the substitution Z~Q#~--I~, Z-iZol ~-~.

Therefore, ~V/Z~, But by eemma 3i ~aE~ , and thus ~ ~rZ~ . Therefore, ~2~

cannot be defined by a system of quasiidentities stable with respect to direct wreaths. The

theorem is proved.

THEOREM 4. The set of quasivarieties of groups, closed with respect to direct wreaths,

which cannot be defined by quasiidentities stable with respect to direct wreaths, has the

cardinality of the continuum.

263

Proof. In Theorem 5 of [3] sequences At,A2,... of nilpotent groups and ~:~I,C~2,.,. of quasiidentities were constructed, ~uch that

a) ~. is false in the group A~

b) ~ is true in the group 2~ for ~j (&,j-<2,...); c) if ~ is true in the torsion-free groups ~ and ~, then ~ is also true in the

group ~ ~ (6=~2,...).

For any subset ~of the set ~ of natural numbers, let ~Z=~A~ (&~[) be the direct

product of the groups ~i (~G[) . From a)-c) and Lemma 2, we see that for f~

Therefore, the set of quasivarieties closed with respect to direct wreaths and contained in

has the cardinality of the continuum. The theorem is proved.

By analogy with the corresponding statement for varieties [6], we can easily prove the

following:

Remark. Let the quasivariety ~ contain a countable set ~IA& ] ~£N 3 of groups such

that A~[Aili~ , dEN3 for any ~e~. Then the quasivariety ~ contains a chain of

quasivarietles which is order-isomorphic to the set of real numbers.

Proof. Let ~ :~--~ be a one-to-one mapping from the set N of natural numbers to

the set Q of rational numbers. With each real number ~ we associate a quasivariety

varieties.

It follows from the above remark that, in the lattice of quasivarieties of groups, we

"very often" come across chains which are order-isomorphic to the set of real numbers. In

the set of groups tieN] in Theor= 4 by an rational particular, by r enumbering numbers,

we easily obtain:

COROLLARY 5. The lattice of quasivarieties of groups contains a chain which is order-

isomorphic to the set of real numbers and, moreover, any quasivariety in this chain is closed

with respect to direct wreaths and is not defined by a system of quasiidentities stable

with respect to direct wreaths.

LITERATURE CITED

i. A. I. Budkin, "Quasiidentities of nilpotent groups and groups with one defining rela- tion," Algebra Logika, 18, No. 2, 127-136 (1979).

2. A. I. Budkln, "The independent axiomatizability of quasivarieties of groups," Mat. Zametkl, 31, No. 6, 817-826 (1982).

3. A. I. Budkin, "Quasivarleties of groups closed with respect to direct wreaths," Mat. Sb., 121, No. 4, 510-522 (1983).

4. A. I. Budkin and V. A. Gorbunov, "On the theory of quasivarieties of algebraic systems," Algebra Logika, 14, No. 2, 373-392 (1975).

5. A. I. Mal'tsev, Algebraic Systems [in Russian], Mir, Moscow (1969). 6. A. Yu. Ol'shanskii, "On certain infinite systems of identities," Tr. Sem. im. I. G.

Petrovsk., ~, 139-146 (1978). 7. K. Chandrasekharan, Introduction to Analytic Number Theory, Springer-Verlag (1968). 8. A. L. Shmel'kin, "On free products of groups," Mat. Sb., 79, No. 4, 616-620 (1969). 9. G. Baumslag, "Groups with one defining relator," J. Austral. Math. Soc., ~, 385-392

(1964). i0. B. B. Newman, "Some results on one-relator groups," Bull. Am. Math. Sot., 74, No, 3,

568-571 (1968).

264