Coset complexes of $p$-subgroups in finite groups

TL;DR

The paper investigates the $p$-local coset poset $_p(G)$ of a finite group $G$ and its geometric and algebraic invariants, aiming to extend Brown's $p$-local perspective to cosets of $p$-subgroups. It develops a robust poset-topology framework: proving homotopy equivalences to intersection posets, deriving a closed-form Euler characteristic $(_p(G))$ via Möbius inversion, and introducing the normalized invariant $p(G)$ which is integer-valued and behaves well under quotients and products. For $p$-closed groups and for products of $p$-TI-groups, precise criteria are given for when $p(G)=1$, including a detailed structure theorem involving $A imes S_3^{m}$ with $m$ even in the $p=2$ case. Additionally, a mod-$p^d$ congruence is established, linking $p(G)$ to the number of Sylow $p$-subgroups and generalizing homological Sylow-type results. Together, these results provide a coherent picture of the homotopy type and Euler characteristic of $p$-local coset posets and their arithmetic consequences in finite group theory.

Abstract

Let $G$ be a finite group and $p$ be a prime. We denote by $C_p(G)$ the poset of all cosets of $p$-subgroups of $G$. We characterize the homotopy type of the geometric realization $|ΔC_p(G)|$ for $p$-closed groups $G$, which is motivated by K.S.Brown's Question. We will further demonstrate that $χ(C_{p}(G)) \equiv |G|_{p'} (\text{mod} p)$ for any finite group $G$ and any prime $p$.

Theorems & Definitions (33)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Corollary 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof