Applying combinatorial results to products of conjugacy classes (Q2197982)

Let \(G\) be a group and \(K\) a finite conjugacy class of \(G\). For a positive integer \(n\), let \(K^n\) be the set of all products with exactly \(n\) factors from \(K\). The author is interested in the increasing sequence \((|K^n|)_n\) and uses combinatorial results to obtain the following Theorem. Suppose \(K=x^G\) is a conjugacy class of \(x\in G\) and \(K\) is finite. Then: (1) if \(|K^2|=\mu |K|\) with \(\mu <3/2\), then \(K^r=x^r[x,G]\) for all \(r\geq2\); (2) if \(|K^3|=(3/2) |K|\), then \(K^r=x^r[x,G]\) for all \(r\geq2\); (3) if \(|K^4|=\mu |K|\) with \(\mu <2\), then \(K^r=x^r[x,G]\) for all \(r\geq4\); (4) if \(|K^5|=2 |K|\), then \(K^r=x^r[x,G]\) for all \(r\geq4\). These results are then used to contribute to conjectures about the solvablility of \(\langle K\rangle\).