UDC 512.542
MSC: 20D60, 05C25
https://doi.org/10.21538/0134-4889-2026-32-1-fon-03
The work is supported by Russian Science Foundation, project 24-11-00119,
https://rscf.ru/en/project/24-11-00119/ .
Dedicated to the bright memory of Professor Otto Helmut Kegel
The spectrum of a finite group $G$ is the set of all element orders of $G$. The Gruenberg—Kegel graph (or the prime graph) $\Gamma(G)$ of a finite group $G$ is defined as follows. The vertex set of $\Gamma(G)$ is the set of all prime divisors of the order of $G$. Two distinct primes $p$ and $q$ are adjacent in $\Gamma(G)$ if and only if there exists an element of order $pq$ in $G$. We say that the problem of recognition by Gruenberg–Kegel graph (by spectrum, respectively) is solved for a finite group if the number of pairwise non-isomorphic finite groups with the same Gruenberg–Kegel graph (spectrum, respectively) as the group under study is known. In 2005, A.V. Vasil’ev completed solving the problem of recognition by spectrum for all finite nonabelian simple groups with orders having prime divisors at most $13$. In this paper we complete the solution of the problem of recognition by Gruenberg–Kegel graph for these groups.
Keywords: finite group, simple group, Gruenberg–Kegel graph (prime graph), recognition
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Received October 23, 2025
Revised November 23, 2025
Accepted November 24, 2025
Published online November 27, 2025
Funding Agency: The work is supported by Russian Science Foundation, project no. 24-11-00119, https://rscf.ru/en/project/24-11-00119/ .
Natalia Vladimirovna Maslova, Dr. Phys.-Math. Sci., Leading Research Fellow, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090, Russia; N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: butterson@mail.ru
Lev Gennadievich Nechitailo, Student, Moscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia, Research Intern, HSE University, 109028, Moscow, Russia,
e-mail: nechitailo.lev@gmail.com
Cite this article as: N.V. Maslova, L.G. Nechitailo. On recognition by Gruenberg–Kegel graph of finite nonabelian simple groups with orders having prime divisors at most 13. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 131–145.
Русский
Н.В. Маслова, Л.Г. Нечитайло. О распознавании по графу Грюнберга–Кегеля конечных неабелевых простых групп, простые делители порядков которых не превосходят 13
Спектр конечной группы $G$ — это множество всех порядков элементов группы $G$. Граф Грюнберга—Кегеля (или граф простых чисел) $\Gamma(G)$ конечной группы G определяется следующим образом. Множество вершин $\Gamma(G)$ — это множество всех простых делителей порядка группы $G$. Два различных простых числа $p$ и $q$ смежны в $\Gamma(G)$ тогда и только тогда, когда в $G$ существует элемент порядка $pq$. Мы говорим, что задача распознавания по графу Грюнберга–Кегеля (соответственно, по спектру) решена для конечной группы, если известно число попарно неизоморфных конечных групп с тем же графом Грюнберга–Кегеля (соответственно, спектром), что и у изучаемой группы. В 2005 году А. В. Васильев завершил решение задачи распознавания по спектру для всех конечных неабелевых простых групп с порядками, имеющими простые делители, не превосходящие $13$. В данной работе завершается решение задачи распознавания по графу Грюнберга–Кегеля для этих групп.
Ключевые слова: конечная группа, простая группа, граф Грюнберга–Кегеля (граф простых чисел), распознаваемость
