On groups whose conjugacy class sizes are not divisible by each other

N. Yang^a, I. B. Gorshkov<sup>b

a</sup> Jiangnan University
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: 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.

MSC: 20D60, 20E45

Received: 21.02.2025
Revised: 06.09.2025
Accepted: 13.09.2025

Language: English

DOI: 10.21538/0134-4889-2025-31-4-300-308

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