Two results on character codegrees

TL;DR

This paper investigates properties of character codegrees in finite groups, establishing bounds related to prime divisors and characterizing prime graphs through graph theory, advancing understanding of group structure via character theory.

Contribution

It provides the best bounds for the size of the prime set of a group under specific codegree prime divisor conditions and characterizes codegree prime graphs using graph theory.

Findings

Bounded the size of the prime set for groups with 2, 3, and 4 prime divisors in codegrees.

Characterized codegree prime graphs for certain groups purely through graph properties.

Abstract

Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of irreducible characters of $G$. The codegree of an irreducible character $\chi$ of the group $G$ is defined as $\mathrm{cod}(\chi)=|G:\mathrm{ker}(\chi)|/\chi(1)$. In this paper, we study two topics related to the character codegrees. Let $\sigma^c(G)$ be the maximal integer $m$ such that there is a member in $\mathrm{cod}(G)$ having $m$ distinct prime divisors, where $\mathrm{cod}(G)=\{\mathrm{cod}(\chi)|\chi\in \mathrm{Irr}(G)\}$. One is related to the codegree version of the Huppert's $\rho$-$\sigma$ conjecture and we obtain the best possible bound for $|\pi(G)|$ under the condition $\sigma^c(G) = 2,3,$ and $4$ respectively. The other is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms.