N.V. Maslova, K.A. Ilenko. On the coincidence of Gruenberg–Kegel graphs of an almost simple group and a nonsolvable Frobenius group ... P. 168-175

Let $G$ be a finite group. Its spectrum $\omega(G)$ is the set of all element orders of $G$. The prime spectrum $\pi(G)$ is the set of all prime divisors of the order of $G$. The Gruenberg—Kegel graph (or the prime graph) $\Gamma(G)$ is a simple graph whose vertex set is $\pi(G)$, and two distinct vertices $p$ and $q$ are adjacent in $\Gamma(G)$ if and only if $pq \in \omega(G)$. The structural Gruenberg—Kegel theorem implies that the class of finite groups with disconnected Gruenberg—Kegel graphs widely generalizes the class of finite Frobenius groups, whose role in finite group theory is absolutely exceptional. The question of coincidence of Gruenberg—Kegel graphs of a finite Frobenius group and of an almost simple group naturally arises. The answer to the question is known in the cases when the Frobenius group is solvable and when the almost simple group coincides with its socle. In this short note we answer the question in the case when the Frobenius group is nonsolvable and the socle of the almost simple group is isomorphic to $PSL_2(q)$ for some $q$.

Keywords: finite group, Gruenberg—Kegel graph (prime graph), nonsolvable Frobenius group, almost simple group

Received January 28, 2022

Revised April 30, 2022

Accepted May 5, 2022

Funding Agency: This work was supported by the Russian Science Foundation (project no. 19-71-10067).

Natalia Vladimirovna Maslova, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Prof., Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: butterson@mail.ru

Kristina Al’bertovna Ilenko, doctoral student, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: christina.ilyenko@yandex.ru

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Cite this article as: N.V. Maslova, K.A. Ilenko. On the coincidence of Gruenberg–Kegel graphs of an almost simple group and a nonsolvable Frobenius group, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 2, pp. 168–175; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 317, Suppl. 1, pp. S130–S135.