MSC: 20D60, 20E45
DOI: 10.21538/0134-4889-2025-31-4-300-308
The work is supported by the Russian Science Foundation (project No. 24-41-10004), https://rscf.ru/project/24-41-10004/ .
Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes excluding $1$. Let us define a directed graph $\Gamma(G)$, the set of vertices of this graph is $N(G)$ and the vertices $x$ and $y$ are connected by an arc from $x$ to $y$ if $x$ divides $y$ and $N(G)$ does not contain a number $z$ different from $x$ and $y$ such that $x$ divides $z$ and $z$ divides $y$. We will call the graph $\Gamma(G)$ the conjugate graph of the group $G$. In this work, we will study finite groups whose conjugate graph is a set of isolated vertices.
Keywords: finite group, conjugacy classes, conjugate graph
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Received February 21, 2025
Revised September 6, 2025
Accepted September 13, 2025
Funding Agency: The work is supported by the Russian Science Foundation (project No. 24-41-10004), https://rscf.ru/project/24-41-10004/.
Nanying Yang, School of Science, Jiangnan University, Wuxi, 214122 China, е-mail: yangny@jiangnan.edu.cn
Ilya Gorshkov, Dr. Phys.-Math. Sci., Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia; Siberian Federal University, Krasnoyarsk, 660041 Russia, е-mail: ilygor8@gmail.com
Cite this article as: Nanying Yang, Ilya Gorshkov. On groups whose conjugacy class sizes are not divisible by each other. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 300–308.
Русский
Н. Янг, И. Горшков. О группах, размеры классов сопряженности которых не делятся друг на друга
Пусть $G$ — конечная группа, и $N(G)$ — множество размеров ее классов сопряженности, за исключением $1$. Определим ориентированный граф $\Gamma(G)$, множество вершин которого совпадает с $N(G)$, а вершины $x$ и $y$ соединены направленным ребром из $x$ в $y$, если $x$ делит $y$ и $N(G)$ не содержит числа $z$, отличного от $x$ и $y$, такого, что $x$ делит $z$, а $z$ делит $y$. Мы будем называть граф $\Gamma(G)$ графом сопряженности группы $G$. В данной работе изучаем конечные группы, граф сопряженности которых представляет собой множество изолированных вершин.
Ключевые слова: конечная группа, классы сопряженности, граф сопряженности
