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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, Volume 28, Number 2, Pages 168–175
DOI: https://doi.org/10.21538/0134-4889-2022-28-2-168-175
(Mi timm1912)
 

On the Coincidence of Gruenberg–Kegel Graphs of an Almost Simple Group and a Nonsolvable Frobenius Group

N. V. Maslovaab, K. A. Il'enkoa

a N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
References:
Abstract: 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 the simple graph with vertex set $\pi(G)$ in which any two vertices $p$ and $q$ are adjacent 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.
Funding agency Grant number
Russian Science Foundation 19-71-10067
This work was supported by the Russian Science Foundation (project no. 19-71-10067).
Received: 28.01.2022
Revised: 30.04.2022
Accepted: 05.05.2022
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2022, Volume 317, Issue 1, Pages S130–S135
DOI: https://doi.org/10.1134/S0081543822030117
Bibliographic databases:
Document Type: Article
UDC: 512.542
MSC: 20D06, 20D60
Language: Russian
Citation: N. V. Maslova, K. A. Il'enko, “On the Coincidence of Gruenberg–Kegel Graphs of an Almost Simple Group and a Nonsolvable Frobenius Group”, Trudy Inst. Mat. i Mekh. UrO RAN, 28, no. 2, 2022, 168–175; Proc. Steklov Inst. Math. (Suppl.), 317, suppl. 1 (2022), S130–S135
Citation in format AMSBIB
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\by N.~V.~Maslova, K.~A.~Il'enko
\paper On the Coincidence of Gruenberg--Kegel Graphs of an Almost Simple Group and a Nonsolvable Frobenius Group
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2022
\vol 28
\issue 2
\pages 168--175
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\crossref{https://doi.org/10.21538/0134-4889-2022-28-2-168-175}
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\jour Proc. Steklov Inst. Math. (Suppl.)
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\vol 317
\issue , suppl. 1
\pages S130--S135
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