Supersolvability and nilpotency in terms of the commuting probability and the average character degree

TL;DR

This paper establishes precise bounds based on the smallest prime dividing a finite group's order to determine when the group is nilpotent or supersolvable, using commuting probability and average character degree.

Contribution

It provides new sharp bounds linking prime divisors, commuting probability, and character degrees to group structure properties.

Findings

Bounds depend on the smallest prime dividing |G|

Conditions guarantee nilpotency or supersolvability

Results are sharp and optimal for given parameters

Abstract

Let $p$ be a prime and let $G$ be a finite group such that the smallest prime that divides $|G|$ is $p$. We find sharp bounds, depending on $p$, for the commuting probability and the average character degree to guarantee that $G$ is nilpotent or supersolvable.