# On the number of irreducible real-valued characters of a finite group

## TL;DR

This paper establishes a bound on the structure of finite groups based on their number of real-valued irreducible characters, and classifies certain groups when this number is five.

## Contribution

It introduces a function bounding the quotient of a finite group by its largest solvable normal subgroup based on real-valued irreducible characters, and classifies groups when this number is five.

## Key findings

- Existence of an integer-valued function f(k) bounding |G/Sol(G)|
- Bound applies when G has at most k real-valued irreducible characters
- Complete classification of G/Sol(G) when k=5

## Abstract

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)| is at most f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k = 5, we further classify G/Sol(G). This partly answers a question of Iwasaki [15] on the relationship between the structure of a finite group and its number of real-valued irreducible characters.