The $A$-Möbius function of a finite group

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 $\mathrm{Aut}(\mathrm{G})$ of a finite group $G$ and denote by ${\mathcal{C}}_A(G)$ the set of $A$-conjugacy classes of subgroups of $G.$ For $H\le G$ let $[H{]}_A=\{H^a\mid a\in A\}$ be the element of ${\mathcal{C}}_A(G)$ containing $H.$ We may define an ordering in ${\mathcal{C}}_A(G)$ in the following way: $[H{]}_A\le [K{]}_A$ if $H^a\le K$ for some $a\in A$. We consider the Möbius function ${\mu }_A$ of the corresponding poset and analyse its properties and possible applications.