On powers of conjugacy classes in finite groups

Abstract

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.