The soluble radical and orbits of certain maps on finite groups

TL;DR

This paper investigates the relationship between the soluble radical of finite groups and the orbits of certain maps defined on the group, providing bounds based on the behavior of these maps on 2-elements.

Contribution

It establishes a bound on the index of the soluble radical in terms of the orbit counts of specific maps on 2-elements, extending previous work by Bray, Wilson, and the authors.

Findings

Bound on the soluble radical's index in terms of orbit counts

Extension of previous results by Bray and Wilson

New connections between group structure and orbit behavior

Abstract

For each element $u$ in a finite group $G$ define a map $\theta_u\colon G\to G$ by $\theta_u(g)=[g^{-u},g]$ and set $\Theta_G(u)=\{g\in G\mid \theta_u^n(g)=g \hbox{ for some } n>0\}$. 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 $u$.