Finite groups with coprime non-linear codegrees

TL;DR

This paper classifies finite groups where the set of non-linear irreducible character codegrees contains multiple elements that are pairwise coprime, revealing structural properties related to character theory.

Contribution

It provides a complete classification of finite groups with multiple coprime non-linear codegrees, a new insight in character theory of finite groups.

Findings

Identifies conditions for groups with coprime non-linear codegrees

Classifies all such finite groups

Establishes relationships between group structure and codegree properties

Abstract

Given a finite group G with an irreducible character \chi \in Irr(G), the codegree of \chi is defined by cod(\chi) = |G :\ker \chi|/\chi(1). The set of non-linear irreducible character codegrees of G is denoted by cod(G|G'). In this note, we classify all finite groups G with |cod(G|G')|> 1 and for each pair of distinct elements m, n \in cod(G|G'), m and n are coprime.