Finite groups in which every irreducible character has either $p'$-degree or $p'$-codegree

TL;DR

This work addresses the problem of classifying finite groups $G$ for which every irreducible complex character $\chi$ satisfies either $\chi(1)$ is coprime to $p$ or $\operatorname{cod}(\chi)$ is coprime to $p$, i.e., $G$ is an $\mathcal{H}_p$-group. The authors develop a structural framework via $N=\mathbf{O}^{p'}(G)$ and $V=\mathbf{O}_{p}(N)$ and split the analysis into $p$-solvable (with $p$-length at most $2$) and non-$p$-solvable cases, employing Itô–Michler, Brauer’s height-zero conjecture, and recent classifications of $p$-exceptional linear groups to constrain the action on $p$-subgroups and irreducible modules. The main contribution is a complete classification of $\mathcal{H}_p$-groups, including explicit structural descriptions in several cases and the introduction of the $\mathcal{H}^*_p$-subclass tied to $p$-defect zero. This advances the understanding of how character codegrees reflect and govern the underlying group structure, with implications for recognizing groups from codegree data and for the interplay between blocks, defects, and group actions on modules.

Abstract

For an irreducible complex character $χ$ of a finite group $G$, the codegree of $χ$ is defined by $|G:\ker(χ)|/χ(1)$, where $\ker(χ)$ is the kernel of $χ$. Given a prime $p$, we provide a classification of finite groups in which every irreducible complex character has either $p'$-degree or $p'$-codegree.

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