The soluble radical and orbits of certain maps on finite groups

For each element $u$ in a finite group $G$ define a map $θ_u\colon G\to G$ by $θ_u(g)=[g^{-u},g]$ and set $Θ_G(u)=\{g\in G\mid θ_u^n(g)=g \hbox{ for some } n>0\}$. Then $θ_u$ induces a permutation of $Θ_G(u)$; let $β_G(u)$ 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 $β_G(u)$ for $2$-elements $u$.

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