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.

Let G be a finite group and ψ(G) = g∈G o(g). There are some results about the relation between ψ(G) and the structure of G. For instance, it is proved that if G is a group of order n and ψ(G) > 211 1617 ψ(C n ), then G is solvable. Herzog et al.in [Herzog et al., Two new criteria for solvability of finite groups, J. Algebra, 2018] put forward the following conjecture:Conjecture. If G is a non-solvable group of order n, thenwith equality if and only if G = A 5 . In particular, this inequality holds for all non-abelian simple groups.In this paper, we prove a modified version of Herzog's Conjecture.2000 Mathematics Subject Classification. 20D60, 20F16.

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