Towards the Classification of Finite Simple Groups with exactly Three or Four Supercharacter Theories

TL;DR

This paper characterizes certain finite simple groups based on the number of supercharacter theories they possess, specifically focusing on those with exactly three or four supercharacter theories.

Contribution

It provides a classification of finite simple groups with precisely three or four supercharacter theories, advancing understanding of supercharacter theory structures.

Findings

Identifies finite simple groups with exactly three supercharacter theories

Identifies finite simple groups with exactly four supercharacter theories

Enhances classification of groups based on supercharacter theory complexity

Abstract

A supercharacter theory for a finite group $G$ is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions.The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.