Character degrees of some $p$-groups

TL;DR

This paper investigates the possible character degrees of certain $p$-groups, especially those with a specific index of the derived subgroup, providing bounds on character degrees in these groups.

Contribution

It establishes restrictions on character degrees of $p$-groups with index $p^2$ of their derived subgroup, including bounds for groups of maximal class.

Findings

Character degrees are restricted for $p$-groups with $|G:G'|=p^2$.

In groups of maximal class, characters of degree greater than $p$ are bounded by $p^{(p+1)/2}$.

Provides new bounds on irreducible character degrees in specific $p$-groups.

Abstract

We restrict the possibilities for the character degrees of $p$-groups $G$ satisfying $|G:G'| = p^2$. E.g. if $G$ is of maximal class and has an irreducible character of degree $> p$, then it has such a character of degree at most $p^{\frac{p+1}{2}}$.