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

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.