Variations on average character degrees and $p$-nilpotence

TL;DR

This paper establishes conditions based on average character degrees that guarantee a solvable group is p-nilpotent, providing new bounds and variations for understanding group structure related to character theory.

Contribution

It introduces new bounds on average character degrees that imply p-nilpotence in solvable groups, expanding the theoretical framework of character-based group analysis.

Findings

If the average degree of p'-characters is below a certain bound, the group is p-nilpotent.

Examples show the bounds are tight and not sufficient for p-nilpotence in all cases.

Multiple variations of the main result are proved, broadening its applicability.

Abstract

We prove that if $p$ is an odd prime, $G$ is a solvable group, and the average value of the irreducible characters of $G$ whose degrees are not divisible by $p$ is strictly less than $2(p+1)/(p+3)$, then $G$ is $p$-nilpotent. We show that there are examples that are not $p$-nilpotent where this bound is met for every prime $p$. We then prove a number of variations of this result.