Huppert's conjecture for character codegrees

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.