Criteria for solvable radical membership via p-elements

TL;DR

The paper generalizes a characterization of the solvable radical in finite groups, showing it suffices to check solvability of generated subgroups with specific p-elements, reducing the verification process.

Contribution

It introduces two new criteria for identifying the solvable radical using only p-elements, simplifying previous methods for finite groups.

Findings

Characterization of the solvable radical via p-elements.

Reduced verification process for solvable radical membership.

Applicable to odd order elements and specific p-elements.

Abstract

Guralnick, Kunyavskii, Plotkin and Shalev have shown that the solvable radical of a finite group $G$ can be characterized as the set of all $x\in G$ such that $<x,y>$ is solvable for all $y\in G$. We prove two generalizations of this result. Firstly, it is enough to check the solvability of $<x,y>$ for every $p$-element $y\in G$ for every odd prime $p$. Secondly, if $x$ has odd order, then it is enough to check the solvability of $<x,y>$ for every 2-element $y\in G$.