The Average of Some Irreducible Character Degrees

TL;DR

This paper establishes bounds on the average degrees of certain irreducible characters in finite groups, which help determine group properties like $p$-nilpotency, considering primes and subfields.

Contribution

It introduces new bounds on the average degrees of irreducible characters not divisible by a prime $p$, depending on $p$ and a chosen subfield $k$, with examples showing optimality.

Findings

Derived bounds depend on prime $p$ and subfield $k$.

Bounds are proven to be optimal with constructed examples.

Results relate character degree averages to group $p$-nilpotency.

Abstract

We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that depends on the prime $p$. If we further restrict our average by fixing a subfield $k$ of the complex numbers and then compute the average of the degrees of the irreducible characters whose degrees are not divisible by $p$ and have values in $k$, then we will see that we obtain a bound that depends on both $p$ and $k$. Moreover, we find examples that make those bounds best possible.