Order evaluation of products of subsets in finite groups and its applications. II

Arad, Zvi; Muzychuk, Mikhail

doi:10.1090/s0002-9947-97-01866-7

Cited by 4 publications

(1 citation statement)

References 8 publications

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“…The same conclusion is attained in [4] when K n , for some n 3, is assumed to be a conjugacy class. Problems about products or powers of conjugacy classes that are the union of two classes have also been addressed (for instance, see also [1] or [2]). In particular, in [4], the authors proposed the following conjecture.…”

Section: Introductionmentioning

confidence: 99%

On powers of conjugacy classes in finite groups

Beltrán

2022

Journal of Group Theory

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Let 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=D\cup D^{-1} for some integer n ≥ 2 n\geq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ \langle K\rangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 n\geq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.

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Section: Introductionmentioning

confidence: 99%

On powers of conjugacy classes in finite groups

Beltrán

2022

Journal of Group Theory

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An Arad and Fisman’s Theorem on Products of Conjugacy Classes Revisited

2022

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A theorem of Z. Arad and E. Fisman establishes that if A and B are two non-trivial conjugacy classes of a finite group G such that either $$AB=A\cup B$$ A B = A ∪ B or $$AB=A^{-1} \cup B$$ A B = A - 1 ∪ B , then G cannot be a non-abelian simple group. We demonstrate that, in fact, $$\langle A\rangle = \langle B\rangle $$ ⟨ A ⟩ = ⟨ B ⟩ is solvable, the elements of A and B are p-elements for some prime p, and $$\langle A\rangle $$ ⟨ A ⟩ is p-nilpotent. Moreover, under the second assumption, it turns out that $$A=B$$ A = B . This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups.

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A structure theorem for small sumsets in nonabelian groups

Serra

Zémor

2013

European Journal of Combinatorics

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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.

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Order evaluation of products of subsets in finite groups and its applications. II

Abstract: Abstract. In this paper we give a new estimate of the cardinality of the product of subsets AB in a finite non-abelian simple group, where A is normal and B is arbitrary. This estimate improves the one given in

Cited by 4 publications

References 8 publications

On powers of conjugacy classes in finite groups

On powers of conjugacy classes in finite groups

An Arad and Fisman’s Theorem on Products of Conjugacy Classes Revisited

A structure theorem for small sumsets in nonabelian groups

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