Partially ordered sets of non-trivial nilpotent π-subgroups II

Abstract

Let G be a finite group, and π a set of prime numbers dividing the order of G. Denote by Nπ(G) and Lπ(G) respectively the totality of non-trivial nilpotent π-subgroups of G, and that of all subgroups U in Nπ(G) such that OπZNG(U)≤U. In this paper, we study homotopy equivalences related to those two posets which are known to have the same homotopy type. As an application, we deal with homology Hn(Nπ(G)) of the associated order complex by making use of Mayer–Vietoris sequences. Furthermore, we provide an algorithm for determining Lπ(GL(n,pe)) where p∉π. The determination of this is eventually reduced to that of irreducible subgroups of GL(n,pe).