DikProAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 7, No. 2, 2006

pp. 265-268

Every infinite group can be generated by
P-small subset

Dikran Dikranjan and Igor Protasov

Abstract. For every infinite group G and every set of generators

S of G, we construct a system of generators in S which is small in the

sense of Prodanov.

2000 AMS Classification: 20D30, 20F05.

Keywords: Group, large set, small set, P-small set.

A subset B of a group G is called large if G = F · B = B · F for some finite
subset F of G. A subset S of a group G is called small if the subset G\F · S · F
is large for every finite subset F of G.

V. Malykhin and R. Moresco [4] posed the following question: can ever infi-
nite group by generated by small subset? This question was answered positively
in [6] (see also [7, Theorem 13.1], some partial results were obtain also in [2]).

Following [2, §2.1] we call a subset S of a group G left small in the sense
of Prodanov (briefly left P-small) if there exist an injective sequence (an)n<ω
such that the family {an · S : n < ω} consists of pairwise disjoint subsets.
Analogously, right small in the sense of Prodanov (briefly right P-small) is
introduced. The set S is called P-small when it is both left P-small and right
P-small. Clearly, all these notions coincide in the abelian case. That was the
case considered by Prodanov [5], who introduced the notion by noticing that if
for a subset A of an abelian group G the difference set A − A is not not large,
then A is P-small.

By [3, Theorem 4.2], every P-small subset of Abelian group is small, but
there are small subsets of Abelian groups which are not P-small. On the other
hand, the free group of rank 2 contains P-small subsets which are not small.
It was proved in [2, Theorem 3.6] that every abelian group has a P-small set
of generators. Furthermore, every free group (more generally, every group ad-
mitting an infinite abelian quotient) and every infinite symmetric group admit



266 D. Dikranjan and I. Protasov

a P-small set of generators [2, Proposition 3.7, Theorem 3.11]. In this paper
we offer a common generalization of all preceding results in our theorem below
by proving that every set of generators of an infinite group contains a P-small
subset of generators.

For a subset A of a group G we denote by 〈A〉 the subgroup generated by
A.

Theorem 1. Let G be an infinite group, A ⊆ G, G = 〈A〉. Then there exists
a small and P-small subset X of G such that 〈X〉 = G and X ⊆ A.

Proof. If G is finitely generated, the statement is trivial since every set of
generators of G contains a finite set of generators. We can take an arbitrary
finite system X, X ⊆ A of generators of G and choose inductively the sequences
(yn)n<ω, (zn)n<ω such that

yn · X ∩ ym · X = ∅, X · zn ∩ X · zm = ∅

for all n, m such that n < m < ω.
Assume that G is not finitely generated and fix some minimal well-ordering

{gα : α < κ} of A ∪ {e}, g0 = e, e is the identity of G. Put G0 = {e} and
x0 = g1. Suppose that, for some ordinal λ < κ, the elements {xα : α < λ}
and the subgroup {Gα : α < λ} have been chosen. If λ is a limit ordinal,
we put Gλ =

⋃
α<λ

Gα, take the first element gβ such that gβ /∈ Gλ and put
xλ = gβ . If λ is a non-limit ordinal, we denote by Gλ the subgroup generated
by Gλ−1 ∪{xλ−1}, take the first element gβ such that gβ /∈ Gλ and put xλ = gβ.
After κ steps we get the subset X = {xα : α < κ} and the properly increasing
chain {Gα : α < κ} of subgroups of G such that X ⊆ A, G = 〈X〉 and
xα ∈ Dα := Gα+1 \ Gα for every α < κ. By [5, Theorem 13.1], X is small.

To show that X is P-small, we build a sequence sequences (yn)n<ω of ele-
ments of G such that

yn · X ∩ yi · X = ∅ (1)

for every i < n. To this end we use the following easy to see properties of the
sets Dα:

(a) G =
⋃

α<κ
Dα is a partition with Dα ∩ Gλ = ∅ whenever λ ≤ α < κ;

(b) Gα · Dα = Dα · Gα = Dα for every α < κ;
(c) |Dm| ≥ |Gm| ≥ 2

m, for all m < ω.

For every m < ω let Xm = {x0, x1, ..., xm}.
Put y0 = e. Suppose that, for some natural number n, the elements

y0, y1, ..., yn−1 have been chosen so that {y0, y1, ..., yn−1} ⊂ Gω and

yi · X ∩ yj · X = ∅

for all i, j such that i < j ≤ n − 1.
To determine yn, we take a natural number m such that {y0, y1, ..., yn−1} ⊂

Gm and

2m > n(m + 1)2.



Every infinite group can be generated by P-small subset 267

By (c) and by the inequality |{y0, y1, ..., yn−1} · Xm · X
−1
m | ≤ n(m + 1)

2 we
can take the element yn ∈ Dm such that

{y0, y1, ..., yn−1} · Xm ∩ yn · Xm = ∅.

By the choice of yn, we have

ynXm ∩ yi · Xm = ∅

for every i < n. If k, l < ω, k > m, then yj xk ∈ Dk for every j ≤ n. Hence
ynxk = yj xl with k, l > m yields k = l and n = j. Now assume that yixk = yjxl
holds with k > m, i, j ≤ n and l ≤ m. Then according to (a) and (b) this is
not possible as yn · xk ∈ Dk, while yj · xl ∈ Gm+1. Analogously, yn · xk = yj · xl
is not possible with k ≤ m and l > m. This proves that

yn · X ∩ yi · X = ∅

for every i < n. After ω steps we get the sequence (yn)n<ω such that the
family {yn · X : n < ω} consists of pairwise disjoint subsets. Applying these
arguments to the set X−1, we get the sequence (zn)n∈ω such that the family
{X · zn : n ∈ ω} consists of pairwise disjoint subsets. Hence, X is P -small. �

Question 2. Let G be an infinite group of cardinality κ. Does there exist a
subset X of G and a κ-sequence (yα)α<κ such that the family {yα · X : α ∈ κ}
consists of pairwise disjoint subsets and G = 〈X〉?

If G is Abelian the answer is positive (see the proof of Theorem 3.6 from
[2]).

Finally, we offer also the following

Question 3.

(a) Let X be a subset of G such that, for every natural number n there exits
a subset Yn of G such that |Yn| = n and the family {y · X : y ∈ Yn} is
disjoint. Is X left P -small ?

(b) By [7, Theorem 12.10], every infinite group can be partitioned into
countably many small subsets. Can every infinite group be partitioned

into countably many P -small subsets?
(c) Let G be an infinite group. Does there exist a system S of generators

of G such that G 6= (S · S−1)n for every natural number n?

Note added in November 2006. Recently T. Banakh and N. Lyaskovska
answered negatively item (a) of Question 3.



268 D. Dikranjan and I. Protasov

References

[1] A. Bella and V. Malykhin, S mall, large and other subsets of a group, Questions and
Answers in General Topology 17 (1967), 183–197.

[2] D. Dikranjan, U. Marconi and R. Moresco, Groups with small set of generators, Applied
General Topology 4 (2) (2003), 327–350.

[3] R. Gusso, Large and small sets with respect to homomorphisms and products of groups,
Applied General Topology 3 (2) (2002), 133–143.

[4] V. Malykhin and R. Moresco, S mall generated groups, Questions and Answers in General
Topology 19 (1) (2001), 47–53.

[5] Iv. Prodanov, S ome minimal group topologies are precompact, Math.Ann. 227 (1977),
117–125.

[6] I. Protasov, E very infinite group can be generated by small subset, in: Third Intern.
Algebraic Conf. in Ukraine, Sumy, (2001), 92–94.

[7] I. Protasov and T. Banakh, Ball Structures and Colorings of Graphs and Groups,
Matem. Stud. Monogr. Series, Vol 11, Lviv, 2003.

Received August 2005

Accepted January 2006

Dikran Dikranjan (dikranja@dimi.uniud.it)
Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienza
206, 33100 Udine, Italy

Igor Protasov (kseniya@profit.net.ua)
Department of Cybernetics, Kyiv National University, Volodimirska 64, Kiev
01033, Ukraine