The Moebius function on the lattice of normal subgroups

TL;DR

This paper explores the lattice structure of normal subgroups in groups, linking the Moebius function to conjugacy classes and irreducible representations, providing new insights into group generation and representation theory.

Contribution

It introduces a novel application of the Moebius function on the lattice of normal subgroups to analyze group generation and faithful irreducible representations.

Findings

Expression for minimal conjugacy classes needed to generate a group

Character table captures the minimal number of conjugacy classes

Moebius function helps extract information on faithful irreducible representations

Abstract

By studying lattices of normal subgroups, especially those of the socle and radical, an expression is obtained for the minimal number of conjugacy classes required to generate a group. This number is shown to be captured by the character table. The Moebius function is then used to extract information on the faithful irreducible representations of a group.