Finite groups of non-prime-power order with exactly four character codegrees
Yu Zeng, Mehdi Ghaffarzadeh, Mohsen Ghasemi, Dongfang Yang
TL;DR
The paper provides a complete characterization of finite groups of non-prime-power order having exactly four irreducible character codegrees, building on a rich body of work on codegrees and Frobenius-type actions. It develops a framework of auxiliary lemmas and analyzes solvable groups with Fitting heights 2 and 3 to derive a finite list of structural possibilities, including specific Frobenius configurations and module-action constraints. The main result, Theorem A, yields a thorough classification for solvable groups and identifies the non-solvable SL2(2^f) family as the remaining case, while also clarifying the relationship between the group’s nilpotent residual and the possible codegree sets. The findings have implications for understanding how spectral properties of characters (via codegrees) constrain group structure and offer precise, checkable families for further study and applications in group theory and representation theory.
Abstract
For an irreducible complex character \(χ\) of a finite group \(G\), the \emph{codegree} of \(χ\) is defined as the ratio \(|G : \ker(χ)| / χ(1)\), where \(\ker(χ)\) represents the kernel of \(χ\). In this paper, we provide a detailed characterization of finite groups of non-prime power order that have exactly four irreducible character co-degrees.