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 C_A(G) the set of A-conjugacy classes of subgroups of G. For H ≤ G let [H]_A = { H^a ∣ a∈ A} be the element of C_A(G) containing H. We may define an ordering in C_A(G) in the following way: [H]_A ≤ [K]_A if H^a ≤ K for some a ∈ A. We consider the Möbius function μ_A of the corresponding poset and analyse its properties and possible applications.