Groups whose prime graph on class sizes has a cut vertex

TL;DR

This paper classifies finite groups based on the structure of their prime graph on class sizes, specifically identifying those groups whose prime graph has a cut vertex, revealing insights into their conjugacy class size distribution.

Contribution

It provides a complete classification of finite groups with a prime graph on class sizes that contains a cut vertex, a novel structural characterization.

Findings

Identifies all finite groups with a prime graph having a cut vertex.

Characterizes the structure of conjugacy class sizes in these groups.

Advances understanding of the relationship between group structure and prime graph properties.

Abstract

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two vertices $p$ and $q$ being adjacent if and only if $pq$ divides some conjugacy class size of $G$. In the present paper, we classify the finite groups $G$ for which $\Delta(G)$ has a cut vertex.