Alternating and Sporadic Simple Groups are Determined by Their Character Degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G) be the set of all irreducible complex character degrees of G forgetting multiplicities, that is, cd(G) = {χ(1) : χ ∈ Irr(G)} and let cd ^*(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be an alternating group of degree at least 5, a sporadic simple group or the Tits group. In this paper, we will show that if G is a non-abelian simple group and cd(G) í cd(H)cd(G)\subseteq cd(H) then G must be isomorphic to H. As a consequence, we show that if G is a finite group with cd^*(G) í cd^*(H)cd^*(G)\subseteq cd^*(H) then G is isomorphic to H. This gives a positive answer to Question 11.8 (a) in (Unsolved problems in group theory: the Kourovka notebook, 16th edn) for alternating groups, sporadic simple groups or the Tits group.