# $p$-groups with exactly four codegrees

## TL;DR

This paper studies $p$-groups with exactly four codegrees, establishing bounds on their nilpotence class and order under various conditions, extending previous results known for groups with fewer codegrees.

## Contribution

It extends the classification of $p$-groups with three codegrees to those with four, providing bounds on class and order under specific structural hypotheses.

## Key findings

- If $G$ has two irreducible character degrees, its nilpotence class is at most 4.
- For groups with coclass at most 3, the order is bounded by $p^7$.
- With additional hypotheses, the order is bounded by $p^{11}$ for coclass at most 7.

## Abstract

Let $G$ be a $p$-group and let $\chi$ be an irreducible character of $G$. The codegree of $\chi$ is given by $|G:\text{ker}(\chi)|/\chi(1)$. Du and Lewis have shown that a $p$-group with exactly three codegrees has nilpotence class at most 2. Here we investigate $p$-groups with exactly four codegrees. If, in addition to having exactly four codegrees, $G$ has two irreducible character degrees, $G$ has largest irreducible character degree $p^2$, $|G:G'|=p^2$, or $G$ has coclass at most 3, then $G$ has nilpotence class at most 4. In the case of coclass at most 3, the order of $G$ is bounded by $p^7$. With an additional hypothesis we can extend this result to $p$-groups with four codegrees and coclass at most 7. In this case the order of $G$ is bounded by $p^{11}$.