On a product of finite subsets in a torsion-free group (original) (raw)
Related papers
On the cardinality of sumsets in torsion-free groups
Bulletin of the London Mathematical Society, 2012
Let A, B be finite subsets of a torsion-free group G. We prove that for every positive integer k there is a c(k) such that if |B| ≥ c(k) then the inequality |AB| ≥ |A| + |B| + k holds unless a left translate of A is contained in a cyclic subgroup. We obtain c(k) < c 0 k 6 for arbitrary torsion-free groups, and c(k) < c 0 k 3 for groups with the unique product property, where c 0 is an absolute constant. We give examples to show that c(k) is at least quadratic in k. To the memory of Yahya Ould Hamidoune
Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2 c and a countable family E of infinite subsets of G, we construct "Baire many" monomorphisms π : G → T c such that π(E) is dense in {y ∈ T c : ny = 0} whenever n ∈ N, E ∈ E , nE = {0} and {x ∈ E: mx = g} is finite for all g ∈ G and m ∈ N \ {0} such that n = mk for some k ∈ N \ {1}. We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of [22, Problem 6.5]. Applications to group actions and discrete flows on T c , Diophantine approximation, Bohr topologies and Bohr compactifications are also provided. Advances in Mathematics 226 (2011) 4776-4795 4777 We refer the reader to [16] for a background on abelian groups. All undefined topological terms can be found in .
A structure theorem for small sumsets in nonabelian groups
European Journal of Combinatorics, 2013
Let G be an arbitrary finite group and let S and T be two subsets such that |S| ≥ 2, |T | ≥ 2, and |T S| ≤ |T | + |S| − 1 ≤ |G| − 2. We show that if |S| ≤ |G| − 4|G| 1/2 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS| ≤ |S| + |H| − 1 or |SH| ≤ |S| + |H| − 1. This extends to the nonabelian case classical reults for abelian groups. When we remove the hypothesis |S| ≤ |G| − 4|G| 1/2 we show the existence of counterexamples to the above characterization whose structure is described precisely.
On a subgroup contained in some words with a bounded length
Discrete Mathematics, 1992
Hamidoune, Y.O., On a subgroup contained in some words with a bounded length, Discrete Mathematics 103 (1992) 171-176. Let G be a group and let A and B be two finite nonvoid subsets of G such that 1$ B. Using results of Kemperman, we show that either IA U B U ABl b IAl + IBI or there exists a nonnull subgroup contained in A U B U Al?. As an application we obtain the following result: Let A,, 4,. . , A, be subsets of a finite group G such that 1 $ A,; 2 c i s k and IA,1 + IA,1 +. . + jAkl 3 ICI. The union of sets of the form A,,A,,. .. Aij; 1 s i, <i, <.. . <ii s k must include a nonnull subgroup. In particular if B is a subset of G\l such that k IBI 2 ICI, the set B U /3* U.. . U B* must contain a nonnull subgroup.
On a combinatorial problem in group theory
Israel journal of mathematics, 1993
Let n be a positive integer or infinity (denote ∞). We denote by W * (n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X 0 ⊆ X, with 2 ≤ |X 0 | ≤ n + 1 and a function f : {0, 1, 2,. .. , k} −→ X0, with f (0) = f (1) and non-zero integers t0, t1,. .. , t k such that [x t 0 0 , x t 1 1 ,. .. , x t k k ] = 1, where xi := f (i), i = 0,. .. , k, and xj ∈ H whenever x t j j ∈ H, for some subgroup H = D x t j j E of G. If the integer k is fixed for every subset X we obtain the class W * k (n). Here we prove that (1) Let G ∈ W * (n), n a positive integer, be a finite group, p > n a prime divisor of the order of G, P a Sylow p-subgroup of G. Then there exists a normal subgroup K of G such that G = P × K. (2) A finitely generated soluble group has the property W * (∞) if and only if it is finite-by-nilpotent. (3) Let G ∈ W * k (∞) be a finitely generated soluble group, then G is finite-by-(nilpotent of k-bounded class).
A Kronecker–Weyl theorem for subsets of abelian groups
Advances in Mathematics, 2011
Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2 c and a countable family E of infinite subsets of G, we construct "Baire many" monomorphisms π : G → T c such that π(E) is dense in {y ∈ T c : ny = 0} whenever n ∈ N, E ∈ E , nE = {0} and {x ∈ E : mx = g} is finite for all g ∈ G and m such that n = mk for some k ∈ N \ {1}. We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of Tkachenko and Yaschenko [23, Problem 6.5]. Applications to group actions and discrete flows on T c , diophantine approximation, Bohr topologies and Bohr compactifications are also provided.
Some problems on splittings of groups. II
Proceedings of the American Mathematical Society, 1987
If G is an additive abelian group, S is a subset of G, M is a set of nonzero integers, and if each element of G\{0} is uniquely expressible in the form ms, where m € M and s € S, then we say that M splits G. A splitting is nonsingular if every element of M is relatively prime to the order of G; otherwise it is singular. In this paper we discuss the singular splittings of cyclic groups of prime power orders and the direct sum of isomorphic copies of groups.
Some results on the partition of groups
Rend. Sem. Math. Univ. Padova 125 (2011), 119 - 146
We investigate some properties of partitions of groups. Some information on the structure of nilpotent groups as well the embedding of groups with nontrivial partitions into a larger one are discussed. Moreover, we consider the structure of groups admitting a nontrivial partition with 111, 222 or 333 subgroups (considering the whole group) with nontrivial partition and the relation between the number of subgroups with nontrivial partition and the number of primes dividing the order of the group. Finally we analyze the relation between the number of components of a nontrivial partition of a group and its order.