Applying combinatorial results to products of conjugacy classes

Abstract

Abstract Let K = x G {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 { | K n | } n ≥ 1 {\{\lvert K^{n}\rvert\}_{n\geq 1}} and consider restrictions on this sequence and the algebraic consequences. In particular, we prove that if | K 2 | < 3 2 ⁢ | K | {\lvert K^{2}\rvert<\frac{3}{2}\lvert K\rvert} or if | K 4 | < 2 ⁢ | K | {\lvert K^{4}\rvert<2\lvert K\rvert} , then K n {K^{n}} is a coset of the normal subgroup [ x , G ] {[x,G]} for all n ≥ 2 {n\geq 2} or 4, respectively. We then use these results to contribute to conjectures about the solubility of 〈 K 〉 {\langle K\rangle} when K n {K^{n}} satisfies certain conditions.