The A-Möbius function of a finite group

Abstract

The Möbius function of the subgroup lattice of a finite group G has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let A be a subgroup of the automorphism group Aut(G) of a finite group G and denote by CA(G) the set of A-conjugacy classes of subgroups of G. For H ≤ G let [H]A = { Ha ∣ a∈ A} be the element of CA(G) containing H. We may define an ordering in CA(G) in the following way: [H]A ≤ [K]A if Ha ≤ K for some a ∈ A. We consider the Möbius function μA of the corresponding poset and analyse its properties and possible applications.