On the character degree graph of solvable groups

Abstract

Abstract. Let G be a finite solvable group, and let ∆(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. A fundamental result by P.P. Pálfy asserts that the complement∆(G) of the graph ∆(G) does not contain any cycle of length 3. In this paper we generalize Pálfy's result, showing that∆(G) does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of ∆(G) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if n is the clique number of ∆(G), then ∆(G) has at most 2n vertices. This confirms a conjecture by Z. Akhlaghi and H.P. Tong-Viet, and provides some evidence for the famous ρ-σ conjecture by B. Huppert.

On the character degree graph of finite groups

“…Therefore, denoting by λ a (non-principal) irreducible constituent of χ U , the degree of χ is divisible by |G : I G (λ)| = |H : I H (λ)|, thus by p 1 · p 3 . As p 4 is not adjacent to p 3 in ∆(G), we conclude that p 4 does not divide χ (1). But this holds for every χ ∈ Irr(G) such that p 1 | χ(1), hence {p 1 , p 4 } ∈ E(G), as desired.…”

On the absence of a normal nonabelian Sylow subgroup

“…The hypotheses of Lemma 4.3 from [6] are satisfied, and therefore ∆(P ′ H) has disconnected components with vertex sets ρ and π * such that (1) |π * | ≥ 2 |ρ| − 1.…”

Bounding the number of vertices in the degree graph of a finite group

“…The latter inequality, for the solvable case, was proved true in [1,Corollary B] as a consequence of the fact that the vertex set of ∆(G) is covered by two suitable subsets of V(G) inducing complete subgraphs. We remark here that the same is not true without the solvability assumption; the smallest example is ∆(PSL 2 (4)), that has three vertices and no edges.…”

“…We remark here that the same is not true without the solvability assumption; the smallest example is ∆(PSL 2 (4)), that has three vertices and no edges. As regards the general case, a recent paper ( [3]) provides some more evidence for Inequality (1), establishing that it holds also for ω(G) = 4. Nevertheless, (1) turns out to be false in general, as shown by the following example.…”

“…Note that Π = {29, 67, 157, 227} is a set of four prime powers (in fact, of four primes) satisfying the above conditions. This provides a counterexample to Inequality (1) in which the number of vertices is 14, whereas the largest clique has size 6.…”