A condition on a certain variety of groups

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

A LIREZA A BDOLLAHI

B ^IJAN T ^AERI

A condition on a certain variety of groups

Rendiconti del Seminario Matematico della Università di Padova, tome 104 (2000), p. 129-134

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ALIREZA ABDOLLAHI

(*) -

BIJAN TAERI

(*)

ABSTRACT - Let a,, ^..., a n be nonzero

integers

^whose greatest ^{common divisor is}

d . We prove that ^an infinite group G is of finite exponent

dividing

^{d if and}

only

if for every n infinite subsets Xl , ... , Xn ^{of G} there exist x, ^E Xl , ^{... , xn E}

Xn

such that

1. Introduction.

Let V be a

variety

^of groups defined

by

^{the law}

v(xl,

^{... ,}

xn)

^{= 1. We}

say that G is a if for _every n infinite subsets

Xl , ... , Xn

^{of G}

there exist

xn E Xn

^{such that}

v(xl,

^{... , = 1.}

In

[4]

^P.S.

Kim,

A.H. Rhemtulla and H. Smith

posed

^the

following question:

^{for which}

variety W,

^is every infinite a

BJ-group?

There exist

positive

^{answers for the}

variety

^C~ of abelian _groups de- fined

by

^{the law}

[x, y]

^{= 1}

(see [7]),

^the

variety C~,2

^defined

by

^{the law}

[x, y]2

^{= 1}

^{(see [5]),}

the varieties

62

^and

63

^of

2-Engel

^and

3-Engel

groups defined

by

^{the laws}

[ x , y , y ] =

^{1 and}

[x, y , y , y ] = 1, respect- ively (see [10]

^and

[11]),

^the

variety

^of groups of

exponent dividing

^n,

defined

by

the law x n ⁼ 1

(see [6]),

^the

variety .Nk

^of

nilpotent

groups of

nilpotency

^{class at most}

k ,

^defined

by

^{the law}

[xl, - - -,

xk

+ 1 ]

^{= 1}

(see [6]),

the

variety

of 3-abelian _groups defined

by

^{the law = 1}

(*) ^Indirizzo

degli

^AA.:

Department

^of Mathematics,

University

^of Isfahan, Isfahan 81744, Iran;

Department

^of Mathematics,

University

^of

Technology

^of Isfahan, Isfahan, ^Iran.

We would like to express ^our

appreciation

^{to our} supervisor ^{Professor A.}

Mohammadi Hassanabadi, ^{who first drew our attention to the}

subject,

^{and his} encouragement ^{and numerous}

helpful

^comments. We also would

sincerely

^{like to}

thank the Referee for

improving

^the

original

result of the paper.

130

(see [1]) and,

^the

variety generated by

^{the law xl ... xn = 1}

(see [3]).

Now let V and W be varieties of _groups defined

by

different laws

... ,

= 1 and _{... ,}

Ym)

⁼

1, respectively.

C. Delizia in

[2]

posed

^the

question

of whether

V# = W# when V = _W,

and for k &#x3E;

1,

^he

investigated

^the

question

^{for the}

varieties V = Vk (the variety

^defined

by

^{the law ... ,}

xl ]

⁼

^I)

^{and ’~ _}

For nonzero

integers

aI, ^{... ,} a n with the

greatest

^{common divisor}

d ,

we have considered the

question

for the varieties V defined

by

^{the law}

~...~=1

^and

83d; proving

THEOREM.

2. Proofs.

LEMMA 1. Let ’V be the

variety defined by

^{the law}

w (xl, - - -, xn ) _

=1.

If

^{G is an}

infinite FC-group

ⁱⁿ

V*,

^{then G E V.}

~ is infinite. Let A be an infinite abelian

subgroup

^of

n Ca(gi).

^{Then A}

by

^{Lemma 3 of}

[3].

Now consider the infinite sets

... ,

gn A . By

^the

property ^BJ

there exist _a1, ... ,

an E A ,

^{such that}

So

and G E V. m

From now on, for nonzero constant

integers

a 1, ... , a n and _any

nonzero

integer

^{a we} denote the varieties defined

by

^{the laws w =}

Xal 1... xn n

^{and w =}

x a , respectively, by

^{’V and}

Ma.

^{Also we denote the}

greatest

^common divisor of a i , ... , a n

by

^d.

LEMMA 2. Let G ^E

B)#

be an

infinite

group.

If

^{the set T}

= ~ g

^E

E G

I gd

⁼

1 ~

^is

^infinite,

^{then G E ’V. In}

particular

^an

infinite

group in

’V-* BV

^{can not have an}

infinite subgroup

^{in V.}

PROOF. Let X be an infinite subset of G. Let

1 ~ j ~

^{n be fixed.}

By considering

^{the infinite subsets}

Xi

⁼

T ,

^{i =}

1, ... , j - 1,

j

⁺

1,

... , n and

Xj

⁼

^X,

^and

^using

^the

property V*,

^we find the ele- ments gi ^E

T ,

^{i =}

1, ... , j - 1, j

⁺

1,

^{... , n and x E} X such that

gn n

^{= 1. But =}

1,

^{so we have Xa) = 1. Thus G E}

e

83t ,

^for

^{all j}

⁼

^1,

^{... , n . Since =}

83a ^{U ff, we}

^{have G e}

83a ,

^for

^{all j}

⁼

=1,

^{... , n . So as}

required.

We define the class

B?( 00 ) including

^all groups G in which _any infinite subset X contains distinct elements _{Xl’ ... , xn} such that

~~...~"=1.

Also,

if k ~ n is an

integer,

^we define the class

’~(k) including

^all groups G in which _any subset X with

I X I ^{= k ,}

^contains distinct elements

... , 1. It is clear

that,

^and

V# c V( oo ).

LEMMA 3. Let G E an

infinite

groups. Then

I

is

finite, for

any g ^E

G ,

^and any

infinite subgroup H of

^{G .}

PROOF. Let H be an infinite

subgroup

^{of G and} suppose, for a contra-

diction,

that there is an

element g

^e

G ,

^such

that H : I

is infinite.

Let T be a

right

transversal to in H . List the elements of T as

tl, t2 ,

^{... under some well}

ordering ~ ,

^so

that ti

^{if i}

j .

Consider the set of all n-element subsets of T . For each t ^e

T(n),

list the elements ... ,

tin

^{of t in}

ascending

^order

given by ~ ,

^{and write}

t

⁼ ... , Create n! + 1 sets, ^one

U a

^{for each}

permutation a

^{of the}

set {1,2, ... , n ~

and V: For each

... , t2n ) ^put

if

and

put t

^{E V if t f1.}

Ua,

^for any ^a.

By Ramsey’s Theorem,

there exists an infinite subset

!ToC7B

^such

that

U,,

^{for some} a, ^or V.

Suppose,

^{for a}

contradiction,

that then the is infinite.

By

^the

property

there exist n distinct elements such that

1,

^{but then}

t = ~ tl ,

^{lies in some}

^Ua,

^{a contra-}

diction.

Thus

To - for

^{some a.}

Moreover, by restricting

the order - to

To

^we may ^assume that

To = ~ tl , t2 , ...I and ti tj

^{if i}

^{j .}

^{Hence for any}

il i2

^...

in,

^{we have}

Now consider a sequence

132

of 2 n + 1 indices and _suppose that

2,

^{... ,}

n ~

^is fixed and let

a(s)

=1~. Define a sequence

jl

^{... as} follows: the first 1~ - 1 ele- ments

ik -1

¹

(if k

^{= 1 this is}

empty), jk

⁼

in + 1,

^and

the last n - k elements

are jk +

=

in+3, ---,in= ^{(if k}

^{= n this is}

empty).

^Thus

Now if we

put jk = in + 2

in the above _sequence, we have

and thus Therefore Now if k

runs

through

^the

set {1,2,

^{... ,}

n ~,

^{so does s . Thus}

n

a contradiction.

It is natural to ask whether _every infinite

)-group belongs

^{to V.}

The _answer, in

general,

^is

negative.

^In

fact,

^{there is an infinite} group

satisfying

the condition for some

positive integer

which does not

belong

^{to W.}

For

example,

^{we can take a} group K

(with

^{m = 2 and n to be a suffi-}

ciently large prime,

n &#x3E;

101° )

^{in Theorem 31.5 in}

[8].

^This

group K

^is

given by

all relations of the form A n 2 - _{1 and A n} ⁼

B. ,

where A and B

run

through

^a

special

^set of words. The

group K

^is

2-generated

^{and of ex-}

ponent n2,

^{and it is a} central extension

of B(2, n) (this

^{is the}

2-generat-

ed free Burnside _group of

exponent n) by

^a

cyclic

^central

subgroup ( z ~

^of

order n.

Thus K is

infinite,

^since

B( 2 , n )

^is

infinite,

^{and K n =}

( z ~

^{is a finite}

(central) cyclic subgroup

^{of K. Now we} show that the set X of all ele- ments of order n in K coincides with

(z~

and thus is finite. For this _pur- pose ^we need to show that all non-central elements of K have order

n 2 .

Take an element x outside

Z(K) = ( z ~.

It follows from the construction of K that x is

conjugate

^{in K to a}

product

where A n = z . Then

= =

1,

^{since k is not divided}

by

^{n .} Thus x is of or-

der

n 2

as

required.

Let,

^{now 0 be the}

variety

^defined

by

^{the law 1. Then}

K E

0(n 2 ). For,

^let xl , ^{... ,} xn2 be distinct elements of K. Thus ... ,

X;:2

are in Kn and so there exists such that

yi

⁼

Therefore

y1 ... yn

^{= c n =}

1,

^since

COROLLARY 4. Let be an

infinite

group, then

is , finite.

PROOF.

By

^Lemma

3, ~

^{G :}

CG ( g d ) ~

^is

^finite,

^for

^{all g}

^{E G. Thus}

^{G d ~}

~ F,

the FC-center of G . If F is infinite then F E

B1, by

^Lemma

1,

^{and so} G

u lfby

^Lemma

^2,

^{and thus}

^{G d}

^{= 1. If F is}

finite,

^then

G d

^is

also

finite,

^as

required.

PROOF OF THE THEOREM. Let G be an infinite _{group in}

B1#

and as- sume

G fI- B1.

^Then

G d

is

finite, by Corollary

^4. Write H : =

CG (G d ).

^Then

G/H is finite and

a d E Z(H),

^for any a E H. We show that

H/FC(H)

^has

exponent

^{2 where}

FC(H)

^is the FC-centre

a 20 FC(H).

Then the set is infinite. Let

Xl = X2 = ... = Xn = X . Then,

^{since G E}

B1#,

there exist _{gl ,} ... ,

gn E H

^{such that}

since Write

i for

^all

i E { 1, ..., n 1.

^Then

Now let then Y is

infinite,

^since

a2f1.FC(H).

^For

there exist such that

Then

a~~"~ ~~=1

and

From

(*)

^and

(**),

^we

get

⁼

1,

^and

arguing similarly

^{with X =}

= X1 = ... = Xi - 1 = Xi + 1 = ... = Xn and Xi = Y, we obtain (ad)Bi = 1 for any

i in the

set {1,

^{... ,}

n 1.

^Then

^{ad = 1,}

^since

B2, ..., P ,) = 1.

^But

from

a d

⁼

_1,

we have

( a g )d

⁼

1,

^for

any g in

^H

and, by

^Lemma

2,

^{the set} is

finite,

^{that is a E}

FC(H),

^a contradiction.

Therefore

H/FC(H)

^{is a}

locally

^finite group,

FC(H)

^{is finite}

by

^Lem-

mas 1 and

2,

^{and so H is}

locally

finite. Then H can not be infinite

by

^Lem-

ma 2. Thus H is finite which forces G to be finite. This is a

contradiction,

which

completes

^the

proof.

134

REFERENCES

[1] A. ABDOLLAHI, ^A characterization

of infinite

^3-abelian groups, Arch. Math.

(Basel), ⁷³ (1999), pp. 104-108.

[2] ^C. DELIZIA, ^On groups with a

nilpotence

^{condition on}

infinite

subsets,

Alge-

bra

Colloq.,

^{2 (1995),} pp. 97-104.

[3] ^G. ENDIMIONI, ^{On a} combinatorial

problem in

^varieties

of

groups, Comm.

Algebra,

²³ (1995), pp. 5297-5307.

[4] ^{P. S.} KIM - A. H. RHEMTULLA - H. SMITH, ^A characterization

of infinite

metabelian groups, Houston J. Math., ¹⁷ (1991), pp. 429-437.

[5] P. LONGOBARDI - M. MAJ, ^A

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^condition

concerning

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^a

given variety

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Ramsey’s

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Algebra,

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Geometry of defining

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Springer-Verlag,

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Manoscritto pervenuto in redazione il 10

maggio

¹⁹⁹⁹