Huppert's conjecture for character codegrees

Abstract

Huppert's ρ‐σ conjecture is one of the main open problems on character degrees of finite groups. A number of ρ‐σ problems have been studied. For instance, T. Keller and J. Zhang considered the ρ‐σ problem for element orders in the 1990s. Recently, a lot of research is being done on character codegrees. Y. Yang and G. Qian studied the ρ‐σ problem for character codegrees in 2017. In this note, we obtain a sharp bound for groups with trivial solvable radical. As a consequence, we improve the general bound of Yang and Qian. For solvable groups, we notice that the ρ‐σ problem for character codegrees is very closely related to the ρ‐σ problem for element orders. In particular, we give a partial negative answer to a speculation by Yang and Qian on the exact bound for the ρ‐σ problem for character codegrees.

Codegrees of primitive characters of solvable groups

“…

We obtain the codegree of a certain primitive character for a finite solvable group, and thereby give a negative answer to a question proposed by Moretó in [8].

…”

“…Motivated by the results in [5,10,11], Moretó considered the converse relation of codegrees and element orders and proposed an interesting question [8,Question B]. He also mentioned that the counterexamples, if exist, seem to be rare.…”

On a Question of Moretó

“…Motivated by the results in [4,7,8], Moretó considered the converse relation of codegrees and element orders and proposed an interesting question [6,Question B]. He also mentioned that counterexamples, if they exist, seem to be rare.…”

A dual version of Huppert's rho-sigma conjecture for character codegrees

“…This concept was introduced in [14]. The analogue of Huppert's ρ-σ conjecture for character codegrees has already been studied in several papers (see, for example, [16,12]). In this note, we consider the dual problem for character codegrees and obtain an almost quadratic bound.…”