Supercharacter theories of dihedral groups

TL;DR

This paper classifies the lattice of supercharacter theories for dihedral groups, extending previous classifications from cyclic groups to more complex dihedral structures, using their cyclic subgroups of rotations.

Contribution

It provides the first complete classification of the supercharacter theory lattice for dihedral groups, linking it to their cyclic subgroups of rotations.

Findings

Classified the supercharacter theory lattice of dihedral groups

Connected the lattice structure to cyclic subgroups of rotations

Extended known results from cyclic to dihedral groups

Abstract

The set of supercharacter theories of a fixed group $G$ forms a natural lattice. An open question in the study of supercharacter theories is to classify this lattice, and to date, this has only been done for the cyclic groups $\mathbb{Z}_n$. In this paper, we classify the supercharacter theory lattice of the dihedral groups $D_{2n}$ in terms of their cyclic subgroups of rotations.