On two M\"obius function for a finite non-solvable group

TL;DR

This paper explores the relationship between two M"obius functions on subgroup lattices in finite groups, focusing on non-solvable groups and identifying cases where known properties do not hold.

Contribution

It investigates the connection between the M"obius functions for non-solvable groups, especially minimal non-solvable groups, and provides examples where the property fails.

Findings

The property holds for solvable groups but not for all non-solvable groups.

Counterexamples are provided, including certain simple groups like M12.

The paper identifies specific classes of non-solvable groups where the property does or does not hold.

Abstract

Let $G$ be a finite group, $\mu$ be the M\"obius function on the subgroup lattice of $G$, and $\lambda$ be the M\"obius function on the poset of conjugacy classes of subgroups of $G$. It was proved by Pahlings that, whenever $G$ is solvable, the property $\mu(H,G)=[N_{G^\prime}(H):G^{\prime}\cap H]\cdot\lambda(H,G)$ holds for any subgroup $H$ of $G$. It is known that this property does not hold in general; for instance it does not hold for every simple groups, the Mathieu group $M_{12}$ being a counterexample. In this paper we investigate the relation between $\mu$ and $\lambda$ for some classes of non-solvable groups; among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property.