Groups with exactly two 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 $\mathrm{Sp}(6,2)$.

7 pages Added work that shows that $\mathrm{Sp}(6,2)$ has only two supercharacter theories