Supercharacter Theories and Semidirect Products

TL;DR

This paper classifies supercharacter theories of semidirect products of Abelian groups and dihedral groups of odd order, introducing homomorphisms of supercharacter theories to relate these structures.

Contribution

It introduces the concept of homomorphisms of supercharacter theories and classifies supercharacter theories of dihedral groups of order 2m for odd m.

Findings

Supercharacter theories of semidirect products are described via those of direct products.

Classification of supercharacter theories of dihedral groups of order 2m for odd m.

Homomorphisms of supercharacter theories facilitate the classification process.

Abstract

We describe the supercharacter theories of the semidirect product of H and K, $H\rtimes K$ in terms of the supercharacter theories of the direct product of H and K in the case when both H and K are Abelian groups. To do this we introduce the concept of a homomorphism of supercharacter theories. This provides a classification of the supercharacter theories of the dihedral groups of order 2m when m is odd using the known classification of the supercharacter theories of cyclic groups.