Groups whose character degree graph has diameter three

Abstract

Let G be a finite group, and let ∆(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. It is well known that, whenever ∆(G) is connected, the diameter of ∆(G) is at most 3. In the present paper, we provide a description of the finite solvable groups for which the diameter of this graph attains the upper bound. This also enables us to confirm a couple of conjectures proposed by M.L. Lewis.

On the character degree graph of solvable groups

“…Proof. As concerns (a), (b) and (c), the statement is a special case of Lemma 3.9 in [2]. Thus we only have to prove (d).…”

“…The fact that, for a finite solvable group G, the set V(G) is covered by two subsets each inducing a clique was already known to be true in three special cases: when ∆(G) is disconnected (as already mentioned), for metanilpotent groups ([5, Theorem A]) and, in the connected case, when the diameter of ∆(G) attains the upper bound 3 (see Remark 4.4 in [2]). By Corollary B, this is indeed a feature of ∆(G) in full generality.…”

On the character degree graph of finite groups

“…Let p 1 , p 2 , p 3 , p 4 be as in the assumptions of part (e). Then (d) yields {p 1 , p 3 } ⊆ π 0 ; actually, setting P i ∈ Syl pi (H) for i = 1, 2, 3, 4, we have that P 1 and P 3 are abelian and normal in H. Moreover, Lemma 2.1 of [7] yields that {p 2 , p 4 } ∩ π 0 = ∅. Let us assume…”

“…Since t is a primitive prime divisor of (p, n), the action of T /K on A is easily seen to be irreducible, and therefore H/K embeds in Γ(A) (see Theorem 2.1 in [15], for instance). As a consequence, denoting by X/K the Fitting subgroup of H/K, by [7, Lemma 2.1] the prime divisors of H/X constitute a clique in ∆(H/K), which is a subgraph of ∆(G); moreover, Lemma 3.7 in [7] yields that X/K is cyclic.…”

K4-free character graphs with seven vertices

“…If we know the structure of ∆(G), we can often say a lot about the structure of the group G. For instance, Casolo et al [3] has proved that if for a finite solvable group G, ∆(G) is connected with diameter 3, then there exists a prime p such that G = P H, with P a normal non-abelian Sylow p-subgroup of G and H a p-complement. For another instance, all finite solvable groups G whose character graph ∆(G) is disconnected have been completely classified by Lewis [7].…”