An Analogue of Huppert’s Conjecture for Character Codegrees

Abstract

Let G be a finite group, let ${\text{Irr}}(G)$ be the set of all irreducible complex characters of G and let $\chi \in {\text{Irr}}(G)$ . Define the codegrees, ${\text{cod}}(\chi ) = |G: {\text{ker}}\chi |/\chi (1)$ and ${\text{cod}}(G) = \{{\text{cod}}(\chi ) \mid \chi \in {\text{Irr}}(G)\} $ . We show that the simple group ${\text{PSL}}(2,q)$ , for a prime power $q>3$ , is uniquely determined by the set of its codegrees.

Nondivisibility Among Irreducible Character Co-Degrees

“…Let q > 3 be a prime power. In [5], the analogue of Huppert's conjecture for character co-degrees has been verified for the simple group PSL 2 (q), that is, if Codeg(G) = CodegPSL 2 (q)), then G PSL 2 (q). We continue this investigation and prove the following result as a corollary of Theorem 1.2.…”

“…Set Codeg(G) = {χ c (1) : χ ∈ Irr(G)} and cd(G) = {χ(1) : χ ∈ Irr(G)}. In [1,3,4,5,8,11], various properties of the co-degrees of irreducible characters of finite groups are studied. By [11, Theorem A], if p | |G|, then p divides some element of Codeg(G).…”

Equivalent version of Huppert's conjecture for $K_3$-groups

“…Also by [7], we see that the conjecture is true for A 5 , A 6 , L 2 (7), L 2 (8), L 2 (17), L 3 (3). Thus in this section we just verify the conjecture for U 3 (3) and U 4 (2).…”

“…In this paper, we are concerned with the following conjecture, inspired by Huppert's conjecture: Conjucture: Let G be a finite group and H a non-abelian simple group. If cod(G) = cod(H), then G ∼ = H. This conjecture first proposed in [7], where the authors verified the above conjecture for all projective special linear groups of degree 2. In this article by using the same method in [7], we verify this conjecture for K 3 -groups.…”

“…If cod(G) = cod(H), then G ∼ = H. This conjecture first proposed in [7], where the authors verified the above conjecture for all projective special linear groups of degree 2. In this article by using the same method in [7], we verify this conjecture for K 3 -groups. Also we recall that Huppert's conjecture is true for K 3 -groups (see [12]).…”