# A characterization of the linear groups ${L}_{2}\left(p\right)$

• Volume: 64, Issue: 2, page 459-464
• ISSN: 0011-4642

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## Abstract

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Let $G$ be a finite group and ${\pi }_{e}\left(G\right)$ be the set of element orders of $G$. Let $k\in {\pi }_{e}\left(G\right)$ and ${m}_{k}$ be the number of elements of order $k$ in $G$. Set $\mathrm{nse}\left(G\right):=\left\{{m}_{k}:k\in {\pi }_{e}\left(G\right)\right\}$. In fact $\mathrm{nse}\left(G\right)$ is the set of sizes of elements with the same order in $G$. In this paper, by $\mathrm{nse}\left(G\right)$ and order, we give a new characterization of finite projective special linear groups ${L}_{2}\left(p\right)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $|G|=|{L}_{2}\left(p\right)|$ and $\mathrm{nse}\left(G\right)$ consists of $1$, ${p}^{2}-1$, $p\left(p+ϵ\right)/2$ and some numbers divisible by $2p$, where $p$ is a prime greater than $3$ with $p\equiv 1$ modulo $4$, then $G\cong {L}_{2}\left(p\right)$.

## How to cite

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Khalili Asboei, Alireza, and Iranmanesh, Ali. "A characterization of the linear groups $L_{2}(p)$." Czechoslovak Mathematical Journal 64.2 (2014): 459-464. <http://eudml.org/doc/262005>.

@article{KhaliliAsboei2014,
abstract = {Let $G$ be a finite group and $\pi _\{e\}(G)$ be the set of element orders of $G$. Let $k \in \pi _\{e\}(G)$ and $m_\{k\}$ be the number of elements of order $k$ in $G$. Set $\{\rm nse\}(G):=\lbrace m_\{k\}\colon k \in \pi _\{e\}(G)\rbrace$. In fact $\{\rm nse\}(G)$ is the set of sizes of elements with the same order in $G$. In this paper, by $\{\rm nse\}(G)$ and order, we give a new characterization of finite projective special linear groups $L_\{2\}(p)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $|G|=|L_\{2\}(p)|$ and $\{\rm nse\}(G)$ consists of $1$, $p^\{2\}-1$, $p(p+\epsilon )/2$ and some numbers divisible by $2p$, where $p$ is a prime greater than $3$ with $p \equiv 1$ modulo $4$, then $G \cong L_\{2\}(p)$.},
author = {Khalili Asboei, Alireza, Iranmanesh, Ali},
journal = {Czechoslovak Mathematical Journal},
keywords = {element order; set of the numbers of elements of the same order; linear group; sets of element orders; sets of numbers of elements of equal orders; projective special linear groups; nse; characterizable groups; order of groups},
language = {eng},
number = {2},
pages = {459-464},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of the linear groups $L_\{2\}(p)$},
url = {http://eudml.org/doc/262005},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Khalili Asboei, Alireza
AU - Iranmanesh, Ali
TI - A characterization of the linear groups $L_{2}(p)$
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 459
EP - 464
AB - Let $G$ be a finite group and $\pi _{e}(G)$ be the set of element orders of $G$. Let $k \in \pi _{e}(G)$ and $m_{k}$ be the number of elements of order $k$ in $G$. Set ${\rm nse}(G):=\lbrace m_{k}\colon k \in \pi _{e}(G)\rbrace$. In fact ${\rm nse}(G)$ is the set of sizes of elements with the same order in $G$. In this paper, by ${\rm nse}(G)$ and order, we give a new characterization of finite projective special linear groups $L_{2}(p)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $|G|=|L_{2}(p)|$ and ${\rm nse}(G)$ consists of $1$, $p^{2}-1$, $p(p+\epsilon )/2$ and some numbers divisible by $2p$, where $p$ is a prime greater than $3$ with $p \equiv 1$ modulo $4$, then $G \cong L_{2}(p)$.
LA - eng
KW - element order; set of the numbers of elements of the same order; linear group; sets of element orders; sets of numbers of elements of equal orders; projective special linear groups; nse; characterizable groups; order of groups
UR - http://eudml.org/doc/262005
ER -

## References

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