Groups whose prime graphs have no triangles

Abstract

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

Prime divisors of irreducible character degrees

“…By Lemma 3.2, r ∈ ρ(N ) and hence r | θ(1) for some θ ∈ Irr(N ). Since σ(M ) ≤ σ(G) ≤ 2 and M/N ∼ = S, by [T,Lemma 4.2] we deduce that θ extends to θ 0 ∈ Irr(M ). Now let λ ∈ Irr(M/N ) with 2 | λ(1).…”

“…By Lemma 3.2, r | θ(1) for some θ ∈ Irr(N ). Since σ(M ) ≤ 2, by applying [T,Lemma 4.2], we deduce that θ extends to θ 0 ∈ Irr(M ). Using Gallagher's Theorem, we must have that σ(M/N ) = 1 and hence M/N ∼ = PSL 2 (4) or PSL 2 (8).…”

“…By Lemma 3.2, we have that τ ⊆ ρ(N ) and since N is solvable, by applying Pálfy's Condition [P, Theorem], there exists ψ ∈ Irr(N ) such that ψ(1) is divisible by two distinct primes in τ. Now by applying [T,Lemma 4.2] again, we obtain a contradiction. This contradiction shows that |ρ(G)| ≤ 2σ(G) + 1 as wanted.…”

“…Let L be a normal subgroup of M C such that L/C ∼ = S. Notice that M C/C ∼ = S 2 . By applying [T,Lemma 4.2], θ extends to θ 0 ∈ Irr(L). By Gallagher's Theorem [I,Corollary 6.17], θ 0 µ ∈ Irr(L) for all µ ∈ Irr(L/C).…”

“…Then π(ϕ(1)) = {2, r} with r > 2. As M C/L ∼ = S, we can apply [T,Lemma 4.2] again to obtain that ϕ extends to ϕ 0 ∈ Irr(M C) and then by applying Gallagher's Theorem,…”