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We show that if p is an odd prime and G is a finite group satisfying the condition that p 2 divides the degree of no irreducible character of G,

Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let \rho(G) be the set of all primes which divide some character degree of G. The prime graph \Delta(G) attached to G is a graph whose vertex set is \rho(G) and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of G. In this paper, we show that if G is a finite group whose prime graph \Delta(G) has no triangles, then \Delta(G) has at most 5 vertices. We also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups. Finally, we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices.Comment: 13 page

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) ⊆ cd(H) then G must be isomorphic to H. As a consequence, we show that if G is a finite group with cd * (G) ⊆ 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.