A characterization of some finite quasisimple groups using their character codegrees

Abstract

Let G be a finite group. For an irreducible character χ of G, define its codegree by cod(χ) = |G : ker χ|/χ(1). Furthermore, define cod(G) = {cod(χ) : χ ∈ Irr(G)}. A recent conjecture of Hung and Moretó states that if cod(G) ⊆ cod(H) and H is a finite non-abelian simple group, then G ≅ H. They verified this conjecture for sporadic groups, alternating groups of degree at least five and many simple groups of Lie type with small Lie rank. We propose an extension of this conjecture as follows: If cod(G) ⊆ cod(H) and H is a finite quasisimple group, then G ≅ H/N where 1 ⩽ N ⩽ Z(G) and cod(G) = cod(H) if and only if G ≅ H. We show that the conjecture holds when H ≅ SL₂(q), q ⩾ 5 or SL₃(q), q ⩾ 2.