Applying combinatorial results to products of conjugacy classes

Abstract

Let {K=x^{G}} be the conjugacy class of an element x of a group G, and suppose K is finite. We study the increasing sequence of natural numbers {\{\lvert K^{n}\rvert\}_{n\geq 1}} and consider restrictions on this sequence and the algebraic consequences. In particular, we prove that if {\lvert K^{2}\rvert<\frac{3}{2}\lvert K\rvert} or if {\lvert K^{4}\rvert<2\lvert K\rvert}, then {K^{n}} is a coset of the normal subgroup {[x,G]} for all {n\geq 2} or 4, respectively. We then use these results to contribute to conjectures about the solubility of {\langle K\rangle} when {K^{n}} satisfies certain conditions.

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

“…So, from Eqs. ( 6) and (7) we conclude that T = T −1 and m = α + β for some β ∈ N * , a contradiction. This contradiction implies that T = ∅, and hence A 2 = A ∪ A −1 .…”

“…The above result provides further evidence of the following conjecture posed in [6]: If A and B are conjugacy classes of a group such that AA −1 = 1 ∪ B ∪ B −1 , then A is solvable. The non-simplicity of G and the solvability of A for some specific cases were obtained in Theorems A and C of [5] and also in Theorem C of [7].…”

On powers of conjugacy classes in finite groups

“…The next theorem is a consequence of the so-called "Freiman inverse problem for < 3 2 " (see [6]), which was referred to in the introduction. This theorem is particularly applied to the square and to the power of a conjugacy class, and actually is a key result in [5].…”

“…So the case jDj D jKj remained unsolved in [4]. Recently, R. D. Camina has shown in [5] that the assertion of the conjecture is true whenever n 4. This is done by using, among others, a useful combinatorial result due to Freiman, which concerns finite subsets of nonabelian groups that have a small doubling.…”

“…This is done by using, among others, a useful combinatorial result due to Freiman, which concerns finite subsets of nonabelian groups that have a small doubling. The purpose of this note is to study the case n D 3 and hence provide a solution to an open question in [5]. We introduce new ideas and techniques, some of which involve general results on products of sets in finite groups; for example, a quite elementary theorem of Arad, Fisman and Muzychuk appeared in [1] (see Theorem 2.1) that provides a lower bound for the cardinality of the product of two normal subsets.…”