Finite groups with few character values

TL;DR

This paper proves that finite groups with fewer than eight character values are solvable, confirming a conjecture, and classifies non-solvable groups with exactly eight character values.

Contribution

It establishes a new solvability criterion based on the number of character values and classifies certain non-solvable groups, extending classical character degree results.

Findings

Groups with fewer than eight character values are solvable

Confirmed Sakurai's conjecture on character values

Classified non-solvable groups with exactly eight character values

Abstract

A classical theorem on character degrees states that if a finite group has fewer than four character degrees, then the group is solvable. We prove a corresponding result on character values by showing that if a finite group has fewer than eight character values in its character table, then the group is solvable. This confirms a conjecture of T. Sakurai. We also classify non-solvable groups with exactly eight character values.