Applying combinatorial results to products of conjugacy classes

AbstractLet {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.