A.S. Kondrat’ev, I.D. Suprunenko, I.V. Khramtsov. On finite 4-primary groups having a disconnected Gruenberg–Kegel graph and a composition factor isomorphic to $L_3(17)$ or $Sp_4(4)$ ... P. 139-155

The Gruenberg–Kegel graph (the prime graph) $\Gamma(G)$ of a finite group $G$ is the graph in which the vertices are the prime divisors of the order of $G$ and two distinct vertices $p$ and $q$ are adjacent if and only if $G$ contains an element of order $pq$. Investigations of finite groups by the properties of their Gruenberg–Kegel graphs form a dynamically developing branch of the finite group theory. A detailed study of the class of finite groups with disconnected Gruenberg–Kegel graphs is one of the important problems in this direction. In 2010–2011, the first and the third authors described the normal structure of finite 3-primary and 4-primary groups with disconnected Gruenberg–Kegel graphs. Unfortunately, the case where a 4-primary group has a composition factor isomorphic to $L_3(17)$ or $Sp_4(4)$  has been omitted in this description. In the present paper, we obtain a description of the groups under consideration in the omitted case. Now a description of the normal structure of finite 4-primary groups with disconnected Gruenberg–Kegel graphs is corrected. In the course of the proof, the 2-modular decomposition matrix of the group $L_3(17)$  is calculated (up to two parameters every of which takes value 1 or 2).

Keywords: finite group, algebraic group, non-solvable 4-primary group, chief factor, disconnected Gruenberg–Kegel graph, character, Brauer character, decomposition matrix

Received November 16, 2021

Revised December 14, 2021

Accepted December 20, 2021

Funding Agency: The first author was supported by the Russian Foundation for Basic Research (project no. № 20-01-00456) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013,  between the Ministry of Education and Science of the Russian Federation and Ural Federal University); the second author was supported by the Institute of Mathematics of the National Academy of Sciences of Belarus (the State Research Programme “Convergence-2025”).

Anatolii Semenovich Kondrat’ev, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: A.S.Kondratiev@imm.uran.ru

Irina Dmitrievna Suprunenko, Dr. Phys.-Math. Sci., Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 200072 Belarus, e-mail: suprunenko@im.bas-net.by

Igor Vladimirovich Khramtsov, Cand. Sci. (Phys.-Math.), OOO “Yandex Tehnology”, Moscow, 119021 Russia, e-mail: ihramtsov@gmail.com

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Cite this article as: A.S. Kondrat’ev, I.D. Suprunenko, I.V. Khramtsov. On finite 4-primary groups having a disconnected Gruenberg–Kegel graph and a composition factor isomorphic to  $L_3(17)$ or $Sp_4(4)$, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 139–155.