The soluble radical and orbits of certain maps on finite groups

Abstract

For each element $u$ in a finite group $G$ define a map ${\theta }_u:G\to G$ by ${\theta }_u(g)=[g^{-u},g]$ and set ${\Theta }_G(u)=\{g\in G\mid {\theta }_u^n(g)=gfor\; somen>0\}$. Then ${\theta }_u$ induces a permutation of ${\Theta }_G(u)$; let ${\beta }_G(u)$ be the number of orbits apart from $\{1\}$. Building on work of J.N. Bray, R.A. Wilson and the second author, we show that the index of the soluble radical of a finite group $G$ is bounded in terms of the values of ${\beta }_G(u)$ for $2$-elements $u$.

18 pages, revised after referee's suggestions