The A-Möbius function of a finite group

2021/09/11 by Volta, Francesca Dalla, Lucchini, Andrea · 1 citation
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2109.05260

Abstract

The Möbius function of the subgroup lettice 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 \rmAut(G) of a finite group G and denote by \mathcal 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 \mathcal C_A(G) containing H. We may define an ordering in \mathcal 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.

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