On the average character degree of some irreducible characters of a finite group

TL;DR

This paper investigates the relationship between the average character degree of certain irreducible characters in finite groups and the solvability of the groups, establishing bounds that guarantee solvability.

Contribution

It proves that if the average character degree of irreducible characters in Irr(G|N) is at most 16/5, then N is solvable, and if strictly less, then G is solvable, with sharp bounds.

Findings

N is solvable if average degree ≤ 16/5

G is solvable if average degree < 16/5

Bounds are proven to be sharp

Abstract

Let G be a finite group and N be a non-trivial normal subgroup of G, such that the average character degree of irreducible characters in Irr(G|N) is less than or equal to 16=5. Then we prove that N is solvable. Also, we prove the solvability of G, by assuming that the average character degree of irreducible characters in Irr(G|N) is strictly less than 16=5. We show that the bounds are sharp.