Forum Problem Lists Definitions FAQ

20.78 (2022)

Open

For an irreducible complex character $\chi$ of a finite group $G$, the codegree of $\chi$ is defined by $\text{cod}(\chi) = |G : \text{ker}\,\chi|/\chi(1)$. Let $\text{Cod}(G)$ be the set of irreducible character codegrees of $G$.

Conjecture: If $G$ has an element of order $m$, then $m$ divides some member of $\text{Cod}(G)$.

Contributor: G. Qian

Progress

The conjecture is proved when $m$ is a prime power (G. Qian, Arch. Math., 97 (2011), 99–103); when $m$ is square-free (I. M. Isaacs, Arch. Math., 97 (2011), 499–501); or when $G$ is solvable (G. Qian, Bull. London Math. Soc., 53 (2021) 820–824); or when $G$ is a symmetric or alternating group (E. Giannelli (J. Algebra Appl., 23, no. 9 (2024), article ID 2450144); or when $G$ is an almost simple group (S. Y. Madanha, Commun. Algebra, 51, no. 7 (2023), 3143–3151); or when $F(G) = 1$ (Z. Akhlaghi, E. Pacifici, L. Sanus, J. Algebra, 644 (2024), 428–441).

Comments

All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic. If you want to submit a proof (or a partial proof), please use the dedicated proof submission form rather than posting it in the comments.
Order by newest first or oldest first.

No comments yet. Be the first to comment.

Proof claims

Proof claims are the responsibility of the submitter. Appearance here does not mean the claim has been checked for mathematical correctness. Moderators only screen for spam, abuse, and obviously low-effort submissions.

No proof claims yet.