Prime simplicial complexes of finite groups

TL;DR

This paper investigates the properties of prime simplicial complexes of finite groups, establishing new recognisability results, providing examples, and classifying groups based on the purity of their complexes.

Contribution

It introduces new recognisability results for prime simplicial complexes, including the first example of a group distinguishable by its complex and spectrum but not by its prime graph, and classifies groups with pure complexes.

Findings

Recognisability by the prime simplicial complex is stronger than by the prime graph.

First example of a group recognisable by its complex and spectrum but not by its prime graph.

Classification of groups with pure prime simplicial complexes.

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.