Further solvable analogues of the Baer-Suzuki theorem and generation of nonsolvable groups

TL;DR

This paper extends the Baer-Suzuki theorem by demonstrating conditions under which elements of certain orders in almost simple groups generate nonsolvable subgroups, providing explicit exceptions and new generation results.

Contribution

It introduces new solvable analogues of the Baer-Suzuki theorem for elements of prime order, involutions, and orders 6 or 9 in almost simple groups, with explicit exception cases.

Findings

Elements of prime order ≥ 5 can generate nonsolvable groups with an involution.

Three conjugates of an involution typically generate a nonsolvable group, except for specific exceptions.

Two conjugates of elements of order 6 or 9 generate nonsolvable groups.

Abstract

Let $G$ be an almost simple group. We prove that if $x \in G$ has prime order $p \ge 5$, then there exists an involution $y$ such that $<x,y>$ is not solvable. Also, if $x$ is an involution then there exist three conjugates of $x$ that generate a nonsolvable group, unless $x$ belongs to a short list of exceptions, which are described explicitly. We also prove that if $x$ has order $6$ or $9$, then there exists two conjugates that generate a nonsolvable group.