Variations on average character degrees and $p$-nilpotence

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.

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