Solvable groups whose degree graphs have two connected components (Q2747313)

The degree graph \(\Delta(G)\) of a finite group \(G\) has vertex set that consists of primes \(p\) with \(p\mid\chi(1)\) for some \(\chi\in\text{Irr}_\mathbb{C}(G)\), and there is an edge between \(p\) and \(q\) if \(pq\mid\chi(1)\) for some \(\chi\in\text{Irr}_\mathbb{C}(G)\). By a result of Manz, Willems and Wolf the degree graph \(\Delta(G)\) has at most three connected components for \(G\) arbitrary and at most two in case \(G\) is solvable.NEWLINENEWLINENEWLINEIn this interesting paper the author completely classifies solvable groups \(G\) for which \(\Delta(G)\) has two connected components.NEWLINENEWLINENEWLINEThe groups which occur fall into six families. They are mainly related to \(\text{SL}(2,3)\); \(\text{GL}(2,3)\); \(C_3\times C_3\) acted on by \(\text{SL}(2,3)\) or \(\text{GL}(2,3)\); finite fields acted on by a semi-linear affine group as \(S_4\) for example, and a further class involving Frobenius groups. For a detailed description we must refer to the original paper. Based on the classification list applications are given (for instance concerning the Fitting height and the derived length). The author mentions an earlier interesting result of Pálfy which states that in case \(\Delta(G)\) has two connected components and \(G\) is solvable the larger component has at least \(2^a-1\) vertices if the smaller one has \(a\) vertices.