Prime simplicial complexes of finite groups

Abstract

The prime simplicial complex $\Pi(G)$ of a finite group $G$ is composed of all sets of primes $S$ where $G$ has an element of order the product of primes in $S$, with the subsets partially ordered by inclusion. This complex was introduced by Peter Cameron as the generalisation of the well-studied prime (or Gruenberg-Kegel) graphs. In this paper, we establish new results concerning two key properties of $\Pi(G)$: recognisability and purity. We demonstrate that recognisability by the prime simplicial complex is strictly stronger than recognisability by the prime graph. Notably, we present the first known example of a group that is recognisable by its prime simplicial complex and spectrum, but not by its prime graph, and is not a direct product of two isomorphic simple groups. Furthermore, we classify groups with pure prime simplicial complexes (i.e., all maximal simplices have the same size) across several infinite families of finite simple groups. We also provide a partial classification for non-abelian finite simple groups whose prime simplicial complex has maximal simplices of size at most 2.