The Average of Some Irreducible Character Degrees

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.