Groups whose degree graph has three independent vertices

Abstract

Let G be a finite group, and let cd(G) denote the set of degrees of the irreducible complex characters of G. This paper is a contribution to the study of the degree graph of G, that is, the prime graph built on the set cd(G). Namely, we characterize finite groups whose degree graph has three independent vertices (i.e., three vertices that are pairwise non-adjacent). Our result turns out to be a generalization of several previously-known theorems concerning the structure of the degree graph.

On the character degree graph of finite groups

“…But L 2 (7) has non-cyclic subgroups of order 21, a contradiction (subgroups of order pq of a Frobenius complement are cyclic). Thus, we conclude that H ≃ L 2 (8). Proof.…”

“…In other words, if λ ∈ M \ {1 M }, then I H (λ) contains both a Sylow t 2 -subgroup and a Sylow t 3subgroup of H as normal subgroups, and these Sylow subgroups are abelian; in particular, H has abelian Hall {t 2 , t 3 }-subgroups, and the same clearly holds for every normal section of H. Take now a minimal normal subgroup N of H which is simple and such that {t 2 , t 3 } ⊆ π(|H : C H (N )|), as in the third paragraph of this proof; we have that H/C H (N ) is an almost-simple group with abelian Hall {t 2 , t 3 }-subgroups, such that t 2 and t 3 are nonadjacent vertices of ∆(H/C H (N )). This contradicts Proposition 2.8(b) of [8], and the desired conclusion follows.…”

“…Let Q be a Sylow 2-subgroup of H. Observe that n 2 (S) = |Syl 2 (S)| divides n 2 (H) = |H : N H (Q)| = (2 6 − 1)/(2 b − 1) ∈ {9, 21, 63}. But for S in the above list of non-abelian simple sections of L 6 (2), we check that the only groups S that satisfy the condition |Syl 2 (S)| divides one among {9, 21, 63} are L 2 (7) and L 2 (8).…”

“…(In the statement above, O π ′ (G) denotes the smallest normal subgroup of G whose index in G is a π ′ -number.) We remark that, in [8], the same class of groups is characterized by the stronger property that ∆(G) contains a triangle, that is, ∆(G) violates Palfy's "three-vertex condition". Therefore, Theorem A is an improvement of the main result of that paper, which in turn generalizes several previously known results concerning the character degree graph (see [8,Corollaries B,C,D]).…”

“…We observe that GL 6 (2) has a unique conjugacy class of maximal subgroups (isomorphic to GL 2 (8) : C 3 = (C 7 × L 2 (8)) : C 3 ) having a section isomorphic to L 2 (8) and that if H is a subgroup of GL 6 (2) having a normal subgroup isomorphic to SL 2 (8), then (H, V ) ∈ N q , where V is the natural module of GL 6 (2).…”