Order complexes of coset posets of finite groups are not contractible

Abstract

We show that the order complex of the poset of all cosets of all proper subgroups of a finite group $G$ is never ${\mathbb{F}}_2$-acyclic and therefore never contractible. This settles a question of K. S. Brown.

16 pages. v2 fixes minor errors

References in corpus (1)

• Divisibility of binomial coefficients and generation of alternating groups

Cited by in corpus (10)

• Divisibility of binomial coefficients and generation of alternating groups
• Between buildings and free factor complexes: A Cohen-Macaulay complex for Out(RAAGs)
• On Boolean intervals of finite groups
• Ore's theorem on subfactor planar algebras
• Higher generating subgroups and Cohen-Macaulay complexes
• Euler totient of subfactor planar algebras
• On invariable generation of alternating groups by elements of prime and prime power order
• Invariable generation of finite simple groups and rational homology of coset posets
• Boolean lattices in finite alternating and symmetric groups
• Subgroup graph methods for presentations of finitely generated groups and the connectivity of associated simplicial complexes