Overgroup coset complexes in finite groups

Abstract

Let G be a finite group and X be a subgroup of G. We investigate the topological properties of the poset CX(G) of cosets Hx in G with X≤H<G. We show that CX(G) is non-contractible if G is solvable or NG(X) contains a Sylow 2-subgroup and a Sylow 3-subgroup of G. This result follows J. Shareshian and R. Woodroofe's work in Adv. Math(2016). We also give some divisibility properties of the Euler characteristic of CX(G) when X is a p-group, which follows K. S. Brown's classical result in J.Algebra(2000).