Independence and strong independence complexes of finite groups

TL;DR

This paper investigates independence complexes on finite groups, introducing $\Sigma(G)$ and the strong version $\tilde{\Sigma}(G)$ and relating them to the power and enhanced power graphs. It establishes that for finite abelian groups, the independence complex determines the subgroup lattice up to order-preserving isomorphism, leading to a full classification of when independence and strong independence coincide. It develops invariant tools for comparing complexes and shows how subgroup lattices influence (and sometimes fail to determine) the complexes, including negative results via Baer-type constructions and explicit $p$-group examples. The authors provide a complete description of when $\Sigma(G)=\tilde{\Sigma}(G)$: nilpotent cases correspond to monotone groups with the basis property, while non-nilpotent cases are Frobenius-type semidirect products with a normal abelian $p$-subgroup and a cyclic $q$-subgroup, under precise action constraints. Overall, the work connects combinatorial invariants of independence complexes to deep group-structural properties, enabling group recognition from these complexes and clarifying when the two notions of independence align.

Abstract

Let $G$ be a finite group. In 2024, Cameron introduced two different concepts of independence (namely independence and strong independence) for the subsets of $G$, yielding to the definition of two simplicial complexes whose vertices are the elements of $G$. The strong independence complex $\tildeΣ(G)$ turns out to be a subcomplex of the independence complex $Σ(G)$. We discuss several invariant properties related to these complexes and ask a number of questions inspired by our results and the examples we construct. We study then the particular case of complexes on finite abelian groups, giving a characterization of the finite groups realizing them. In conclusion, answering a question of Cameron, we classify all finite groups in which the two concepts of independence coincide.

Theorems & Definitions (92)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• Lemma 2.5
• Proposition 2.6
• Example 2.7