Recognizing $\mathrm{PSL}(2,p)$ in the non-Frattini chief factors of finite groups

TL;DR

This paper investigates the structure of finite groups with specific probabilistic generation properties, showing that if two groups share the same generation probability and have certain nonabelian composition factors, then their non-Frattini chief factors are identical.

Contribution

It establishes a criterion to identify non-Frattini chief factors in finite groups based on probabilistic generation data, focusing on groups with $ ext{PSL}(2,p)$ factors for non-Mersenne primes.

Findings

Groups with the same generation probability and $ ext{PSL}(2,p)$ factors have identical non-Frattini chief factors.

The result applies specifically to primes $p eq 2^k - 1$, i.e., non-Mersenne primes.

Provides a structural link between probabilistic generation and group composition factors.

Abstract

Given a finite group $G$, let $P_G(s)$ be the probability that $s$ randomly chosen elements generate $G$, and let $H$ be a finite group with $P_G(s)=P_H(s)$. We show that if the nonabelian composition factors of $G$ and $H$ are $\mathrm{PSL}(2,p)$ for some non-Mersense prime $p\geq 5$, then $G$ and $H$ have the same non-Frattini chief factors.