Finite groups whose prime graph on class sizes is a block square

TL;DR

This paper characterizes the structure of finite groups whose prime graph on conjugacy class sizes forms a block square, revealing specific structural properties of such groups.

Contribution

It provides a complete classification of finite groups with a prime graph on class sizes that is a block square, a novel structural insight.

Findings

Prime graph on class sizes forms a block square for certain groups

Structural characterization of these groups is achieved

New connections between group structure and prime graph shape

Abstract

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, and two distinct vertices $p, q$ are adjacent if and only if $pq$ divides some class size of $G$. In this paper, we characterise the structure of those groups $G$ whose prime graph $\Delta(G)$ is a block square.