Character degree sums in finite nonsolvable groups

TL;DR

This paper proves the existence of a significant irreducible character in minimal normal nonabelian subgroups of finite groups, resolving two open questions in the field.

Contribution

It establishes the existence of a nontrivial irreducible character of degree at least 5 that extends to the whole group, addressing open problems.

Findings

Existence of an irreducible character of degree ≥ 5 in minimal normal nonabelian subgroups

Extension of this character to the entire group G

Resolution of two open questions in group theory

Abstract

Let N be a minimal normal nonabelian subgroup of a finite group G. We will show that there exists a nontrivial irreducible character of N of degree at least 5 which is extendible to G. This result will be used to settle two open questions raised by Berkovich and Mann, and Berkovich and Zhmud'.