The combinatorics and the homology of the poset of subgroups of p-power index

Abstract

AbstractFor a finite group G and a prime p the poset Sp (G) of all subgroups H ≠ G of p-power index is studied. The Möbius number of the poset is given and the homotopy type of the poset is determined as a wedge of spheres. We describe the representation of G on the homology groups of the order complex of Sp (G) and show that this representation can be realized by matrices with entries in the set {+1, -1, 0}. Finally a CL-shellable subposet of Sp (G) is exhibited for odd primes p