The homotopy types of the posets of p-subgroups of a finite group

TL;DR

This paper investigates the homotopy characteristics of posets of p-subgroups in finite groups, clarifying their topological properties and algebraic conditions affecting contractibility.

Contribution

It provides a negative answer to a longstanding question about the relationship between the contractibility of two posets and characterizes when Ap(G) is contractible based on group properties.

Findings

Negative answer to Stong's question on contractibility

Characterization of Ap(G) contractibility via algebraic properties

Insight into homotopy types of p-subgroup posets

Abstract

We study the homotopy properties of the posets of p-subgroups Sp(G) and Ap(G) of a finite group G, viewed as finite topological spaces. We answer a question raised by R.E. Stong in 1984 about the relationship between the contractibility of the finite space Ap(G) and that of Sp(G) negatively, and describe the contractibility of Ap(G) in terms of algebraic properties of the group G.