On the coincidence of Gruenberg-Kegel graphs of an almost simple group and a nonsolvable Frobenius group
Let \(G\) be a finite group and let \(\pi(G)\) be the set of prime divisors of \(|G|\). The Gruenberg-Kegel graph (or the prime graph) \(GK(G)\) is constructed as follows: the vertices of \(GK(G)\) are elements of the set \(\pi(G)\) and two vertices \(p,q\in\pi(G)\) are joined by an edge if and only if \(G\) contains an element of order \(pq\). This notion was introduced by Gruenberg and Kegel in the 1970s. One of the first results concerning prime graphs of groups was the unpublished Gruenberg-Kegel theorem, which described the structure of a finite group whose prime graph is disconnected. This theorem was published later in the paper by \textit{J. S. Williams} [J. Algebra 69, 487--513 (1981; Zbl 0471.20013)]. Frobenius groups and \(2\)-Frobenius groups play an important role in characterizing the structure of finite groups with the disconnected Gruenberg-Kegel graph. In this connection, in [Proc. Steklov Inst. Math. 283, S139--S145 (2013; Zbl 1307.20023); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 18, No. 3, 99--105 (2012)] \textit{M. R. Zinov'eva} and \textit{V. D. Mazurov} stated the following question: which finite simple groups have the same Gruenberg-Kegel graph as Frobenius groups or \(2\)-Frobenius groups? In the same paper they found all finite simple groups having the same Gruenberg-Kegel graph as a Frobenius group or a \(2\)-Frobenius group. In the paper under review, the authors find all finite almost simple groups with the socle isomorphic to \(\mathrm{PSL}_2(q)\) for which there exists an nonsolvable Frobenius group with the same Gruenberg-Kegel graph. There are \(7\) such groups.
scientific article; zbMATH DE number 7618863
scientific article