The M\"obius Number of the Socle of any Group

TL;DR

This paper derives a formula for calculating the M"obius number of the socle of any group, especially focusing on direct products of simple groups, linking it to the M"obius numbers of the constituent simple groups.

Contribution

It provides a new explicit formula for the M"obius number of the socle of any group, particularly for direct products of simple groups, connecting it to known M"obius numbers.

Findings

Formula for M"obius number of the socle of any group.

Explicit computation for direct products of simple groups.

Connection between M"obius numbers of simple groups and their products.

Abstract

We obtain a formula for the M\"obius number of the socle of any group. Specifically, the M\"obius number of a direct product of simple groups is computed in terms of the M\"obius numbers of the simple groups which make up the product.