Now we present a reformulation of the class A. Let X and Y be two infinite subsets of a group G. Then it is clear that XY n Y X i- 0 if and only if IG E X-l y-l XY.
Bulletin Vol.
of the
24, No.2,
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Ira.nian
Ma.thematical
pp 41-47
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Society
(1998)
A CHARACTERIZATION OF INFINITE ABELIAN GROUPS
Alireza Abdollahi
and Aliakbar
Department University
Mohammadi
Hassanabadi
of Mathematics
of Isfahan, Isfahan, Iran
In this note we prove that J.n infinite group Gis abelian if and only if for every two infinite subsets X and Y of G, 0,
where AB
=
{ab I a E A, bE B} for A, B ~
XY n YX f.
G.
1.Introd uction Many properties equivalent to commutativity
have been considered by
various people, for example in [6] Lennox, Mohammadi Hassanabadi and Wiegold introduced
the class P~-groups,
where n is a positive integer,
as follows: G E P~ if and only if every infinite set of n-sets in G contains a pair X, Y of different members such that XY = Y X, where AB
{ab I a E A , bE B} for A, B ~ o Received:
1991 AMS
=
G. In [6] it is shown that infinite groups
December 28 1998
subject classification:
f( ey words and phrases: Research supported
20F99.
infinite, groups, commutativity.
by the National
tional Research Project
Research Council of L R. Iran (NRC) as a Na-
under the grant number 3618.
o
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Alireza Abdollahi
42
--
-
-
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---
----
and Aliakbar Mohammadi
-
Hassanabadi I
in p~
are abelian if n
=
2 or n
=
3. This result was later extended
in
> 1 then every infinite P~-group
is
[5], where it was shown that if n
abelian. Herc we consider another charactcrization
of infinitc abelian groups. Wc
say that a group G is an A-group or satisfies the property if for every two infinite subsets X and Y of G, XY
A if
and only
i-
0. Our
nYX
main result is the following. Theorem.
Every infinite A-group is abelian.
A. Let X and Y be two infinite subsets of a group G. Then it is clear that XY n Y X i- 0 Now we present
a reformulation
of the class
if and only if IG E X-l y-l XY.
Therefore
G E
A
every two infinite subsets X and Y of G, IG E X-ly-l variety of abelian
groups is generated
above observation
leads us to the following definitions.
Let w El, ••.
,Et
= x~
if and only if for
XY.
by the law x-ly-lxy
... x~: be a word in the free group of rank n
E {-1,1}.
Suppose that
Since the
=
1, the
> 0, where
G is a group and Xl,,,,,Xn
are
subsets of G, we define
Let V(w) be the variety generated
by the law W(Xl,
... ,Xn)
=
1. We
denote by V( w*) and V( w) the class of groups G satisfying the following conditions, respectively: G E V(w*) whenever Xl"",Xn
are infinite subsets of G, there exist
Xl"",Xn E Xn such that W(Xl, ... ,Xn) = 1. G E V( w) whenever 1G E w(Xl, ... , X;..) for all infinite subsets Xh"" Xl
E
Xn
ofG. Clearly, we have :F U V( w) ~ nite groups.
Apparently,
V( w*) ~
V( w), :F being the class of fi-
there is no example
of an infinite group in
D
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"'"
A characterization of infinite abelian groups
43
V( w*) which does not belong to V( w). Note that for certain words w, the equality :F U V( w) = V( w*) is known: it is the case with w = xm, W = [x, y] = x-Iy-Ixy and more generally w = [Xl,"" Xn] [11]. This equality is also proved for w = [x, yJ2 [8], w = [x, y, y] [17], w = [x, y, y, y] [18], w = Xl' .• Xn [3] and w = (XIX2)3(xfx~)-1 [2]. The origin of these results is a question of similar nature
of P. Erdos [12], since this first paper,
problems
have been the object of several articles, (for example
[1],[4] ,[7] ,[8] ,[1 0] ,[13] ,[14]). The class V( w*) have been firstly introduced
in [4] and [11]. As far as
we know the class V( ill) has not been considered in the literature
so far.
2. Proofs In order to establish
our main result, we need two lemmas, the first of
which is the key to the proof of the Theorem. Let G be an infinite A-group.
Lemma 2.1.
Then GG(x) is infinite
for all x in G. Proof. T
Assume that
= G\J(.
J(
=
Then T is infinite.
T. Therefore
IG : GG(x)1
{x E GIG G( x) is infinte } =/: G and let For otherwise, for x E T, {xu
Ig
is finite and so x E J(, a contradiction.
E G} ~
Thus
T is infinite.
{x E G I aX = g}. Then either S(a,g) = o or it is a right coset of GG(a) and so is finite if a E T. Suppose, Now, for g,a E G let S(a,g)
for a contradiction,
A
=
{al,a2,""}
=
that T is infinite. We construct and B
=
{bl,b2,
two infinite subsets
} ofT such that AB n BA = 0. ,an} and Bn = {bI, ... ,bn} for
An = {al, all n E N such that AnBn n BnAn = 0. Let n = 1. There exist aI, bl E T such that al b1 =/: bl aI, since otherwise the infinite set T is a
We define by induction
subset of the centralizer Thus we have AIBI
of any element in T, which is a contradiction.
n BIAI
=
0. Now, suppose inductively
that n
>1
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Alireza Abdollahi
44
and
--------
Aliakbar
and we have already defined the subsets An-l
=
n Bn-lAn-l
An-lBn-l
-------
Mohammadi
lIassanabadi
and Bn-l
of T such that
An and Bn, we first define bn as
0. To obtain
follows: Let
M is a finite set and we may choose bn to be any element of the infinite set T\M. Then we define an as follows: Let
Now N is finite and T\N is infinite. We choose an to be any element in the infinite set T\N. considering
Set An
=
u {an} and Bn = Bn-l u {bn}. By
An-l
the following equalities
n
and the method of choosing an and bn, one can easily check that AnBn BnAn
=
Therefore
0.
i, j, k, lEN,
then aibj
An
=
which is a contradiction.
n nA
=
bka{ E AtBt
0, since if aibj
n BtAt where t
= =
bka{ for some max{ i,j, k, I},
Hence G is not a A-group, a contradiction
.
• Lemma 2.2. subgroup
Then every infinite
II of G contains the derived subgroup G' of G. In particular,
G is nilpotent
Proof. {h E H
Let G be an infmite A-group.
of class at most 2.
Let II be an infinite subgroup
I
g-lhg
1-
H}.
If X is infinite,
of G and 9 E G. Let X then (gX)X
n X(gX)
of-
= 0,
o
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A characterization
45
of infinite abclian groups
which is easily seen to lead to a contradiction.
Thus X is finite and
g-J hg E II for all but finitely many elements
h of Ii and hence II is
normal in G. Kow let x, y E G and consider the infmite subsets and yII.
n (yII)(xII)
Then (xII)(yII)
[.r, y] E II. Therefore intersection
xII
=/= 0, from which it follows that
G' lies in II and by Lemma 2.1, G' lies in the
of the centralizers
of all the elements of G and so is in the
center of G; thus G is nilpotent Now we see how the property
of class at most 2 .•
A
will induce the commutativity
of an
infinite group. Proof of the Theorem. G is non-periodic Consider
II
=
Let G be an infinite A-group.
Suppose that
and a is an element of infinite order in G. Let x, y E G.
(a, x, y) which is an infinite finitely generated
"-.::: ====~,.
A-group,
and thus II is residually finite, by Lemma 2.2. Now II is abelian, since by Lemma 2.2, the derived subgroup of II lies in the intersection normal subgroups
of finite index in Ii. Thus xy
= yx
which means that
G is abelian. Therefore we may assume that G is a torsion group. contradiction, there cannot
that G is non-abelian. be a subgroup
are both infinite.
Suppose,
for a
Now observe that by Lemma 2.2
of G of the form U
X
V where U and V
Thus each abclian subgroup of G satisfies the minimal
condition,
by Theorem
condition
(see Theorem
4.3.11 of [16]. Hence G satisfies the minimal 15.2.2 of [16]). Therefore
[16], G is a finite extension a quasi cyclic p-group, of generators
of all
by Theorem
3.14 of
of a central Priifer group D. Thus D is
for some prime p. Now, let al, a2,.·.
be a set
of the quasi cyclic p-group D which satisfy the following
relations:
Now suppose that [x, y] =/= 1 for some x, y E G. By Lemma 2.2, [x, y] E D and so [x, y] is of order
pn
for some n E N. Let X
=
{an+2, an+3, ... }. By
o
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46
Alireza Abdollahi
the property
A, there exist
aj, aj, ak, a/ E X such that
and so [x, y]
=
Thus akalaj1a;1
aka/aj1a;1.
least pn+1, a contradiction.
and Aliakbar Mohammadi
Therefore xy
•
1=
= yx
Hassanabadi
yajxaj
=
xakya/
1 and so is of order at
and the proof is complete .
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Finitely generated soluble groups with an Engel con-
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subsets,
to appear
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[2] A. Abdollahi, A characterization Math. (Basel) 72(1999),
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