From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical

TL;DR

This paper characterizes the solvable radical of finite and linear groups by showing that elements of prime order greater than 3 belong to the radical if and only if they generate solvable subgroups with all their conjugates, leading to a new criterion for group solvability.

Contribution

It provides a sharp, conjugacy-based criterion for identifying elements in the solvable radical of finite and linear groups, extending classical results.

Findings

Elements of prime order > 3 are in the radical iff they generate solvable subgroups with all conjugates.

A group is solvable iff every pair of elements in each conjugacy class generates a solvable subgroup.

The criterion applies to both finite and linear groups, unifying and extending previous characterizations.

Abstract

We prove that an element $g$ of prime order $>3$ belongs to the solvable radical $R(G)$ of a finite (or, more generally, a linear) group if and only if for every $x\in G$ the subgroup generated by $g, xgx^{-1}$ is solvable. This theorem implies that a finite (or a linear) group $G$ is solvable if and only if in each conjugacy class of $G$ every two elements generate a solvable subgroup.