LAUSR

Abstract Let G be a finite group, and π a set of prime numbers dividing the order of G. Denote by N π ( G ) and L π ( G ) respectively the totality of non-trivial nilpotent π-subgroups of G, and that of all subgroups U in N π ( G ) such that O π Z N G ( 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 H n ( N π ( G ) ) of the associated order complex by making use of Mayer–Vietoris sequences. Furthermore, we provide an algorithm for determining L π ( G L ( n , p e ) ) where p ∉ π . The determination of this is eventually reduced to that of irreducible subgroups of G L ( n , p e ) .