Groups with exactly two supercharacter theories

TL;DR

This paper classifies finite groups with exactly two supercharacter theories, identifying specific solvable and nonsolvable groups that meet this criterion.

Contribution

It provides a complete classification of finite groups with exactly two supercharacter theories, including explicit examples of solvable and nonsolvable groups.

Findings

Solvable groups with two supercharacter theories are Z_3 and S_3.

The only nonsolvable group with two supercharacter theories is Sp(6,2).

The classification helps understand the structure of groups based on their supercharacter theories.

Abstract

In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are $\mathbb{Z}_3$ and $S_3$. We also show that the only nonsolvable group with two supercharacter theories is ${\rm Sp} (6,2)$.