The $A$-M\"{o}bius function of a finite group

TL;DR

This paper generalizes the Möbius function of subgroup lattices in finite groups by incorporating automorphism group actions, analyzing its properties and potential applications in group theory.

Contribution

It introduces an $A$-Möbius function based on automorphism group actions, extending classical subgroup lattice Möbius functions and exploring their properties.

Findings

Defined the $A$-conjugacy classes and ordering in subgroup sets.

Analyzed properties of the $A$-Möbius function.

Discussed potential applications in finite group theory.

Abstract

The M\"{o}bius 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 $\rm{Aut}(G)$ of a finite group $G$ and denote by $\mathcal C_A(G)$ the set of $A$-conjugacy classes of subgroups of $G.$ For $H\leq G$ 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: $[H]_A\leq [K]_A$ if $H^a\leq K$ for some $a\in A$. We consider the M\"{o}bius function $\mu_A$ of the corresponding poset and analyse its properties and possible applications.