On the prime graph of simple groups

TL;DR

This paper investigates the prime graphs of finite simple groups, specifically identifying when a simple group and its proper subgroup share identical prime graphs, advancing understanding of their structural properties.

Contribution

It characterizes the pairs of simple groups and their proper subgroups with identical prime graphs, a novel classification in the study of prime graphs of finite groups.

Findings

Identified all pairs (G, H) with G simple and H a proper subgroup such that Γ(G) = Γ(H).

Provided new insights into the relationship between simple groups and their subgroups via prime graphs.

Enhanced understanding of the structure of prime graphs in finite simple groups.

Abstract

Let $G$ be a finite group, let $\pi(G)$ be the set of prime divisors of $|G|$ and let $\Gamma(G)$ be the prime graph of $G$. This graph has vertex set $\pi(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that $\Gamma(G) = \Gamma(H)$.