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Matematicheskie Zametki, 2023, Volume 113, Issue 1, Pages 109–117
DOI: https://doi.org/10.4213/mzm12962
(Mi mzm12962)
 

Influence of Conjugacy Class Sizes of Some Elements on the Structure of a Finite Group

Ruifang Chen, Xianhe Zhao

College of Mathematics and Computer Science, Henan Normal University, P. R. China
References:
Abstract: Let $G$ be a finite group. For every element $x\in G$, the set $\{x^g=g^{-1}xg: g\in G\}$ is called the conjugacy class of $x$ in $G$ and is denoted by $x^G$. The conjugacy class size of $x$ in $G$ is denoted by $|x^G|$ and is equal to $|G:C_G(x)|$. An element $y$ of $G$ is said to be primary or biprimary if the order of $y$ is divisible by exactly one or two distinct primes. For a positive integer $n$ and a prime $p$, if $e>0$ is an integer such that $p^e$ divides $n$ and $p^{e+1}$ does not divide $n$, then $p^e$ is called the $p$-part of $n$. Let $p$ be a prime divisor of $p$ such that $(p-1,|G|)=1$. We prove that $G$ is solvable and $p$-nilpotent if the conjugacy sizes of all noncentral primary and biprimary elements in $G$ have the same $p$-part. On the other hand, suppose that $N$ is a normal subgroup of $G$; we write $\operatorname{cs}_G(N)=\{|x^G|:x\in N\}$. Suppose that $\operatorname{cs}_G(N)=\{1,n_1,n_2,\dots,n_t\}$, where $1<n_1<n_2<\cdots<n_t$. Denote by
$$ M_N(G)=\langle x^G:x\in N,\,x^G=1\text{ or }n_1\rangle $$
a subgroup of $G$. We prove that if $C_G(F(G))\le F(G)$ and $[x,F(G)]$ is a normal subset of $F(G)$ for every $x\in N$ with $|x^G|=1$ or $n_1$, then $M_N(G)$ is a nilpotent group with nilpotency class at most 2.
Keywords: conjugacy class size; solvable group; $p$-nilpotent group; nilpotency class.
Funding agency Grant number
National Natural Science Foundation of China U1504101
11501176
Doctoral Research Foundation of Henan Normal University 5101019170127
This research was supported in part by the National Natural Science Foundation of China under grants U1504101, 11501176 and by the Doctoral Research Foundation of Henan Normal University under grant 5101019170127.
Received: 17.11.2020
Revised: 28.01.2021
English version:
Mathematical Notes, 2023, Volume 113, Issue 1, Pages 109–115
DOI: https://doi.org/10.1134/S000143462301011X
Bibliographic databases:
Document Type: Article
UDC: 512
Language: Russian
Citation: Ruifang Chen, Xianhe Zhao, “Influence of Conjugacy Class Sizes of Some Elements on the Structure of a Finite Group”, Mat. Zametki, 113:1 (2023), 109–117; Math. Notes, 113:1 (2023), 109–115
Citation in format AMSBIB
\Bibitem{RuiXia23}
\by Ruifang Chen, Xianhe Zhao
\paper Influence of Conjugacy Class Sizes of Some Elements on the Structure of a Finite Group
\jour Mat. Zametki
\yr 2023
\vol 113
\issue 1
\pages 109--117
\mathnet{http://mi.mathnet.ru/mzm12962}
\crossref{https://doi.org/10.4213/mzm12962}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4563352}
\transl
\jour Math. Notes
\yr 2023
\vol 113
\issue 1
\pages 109--115
\crossref{https://doi.org/10.1134/S000143462301011X}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85149934092}
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