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Title: A characterization of the linear groups $L_{2}(p)$ (English)
Author: Khalili Asboei, Alireza
Author: Iranmanesh, Ali
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 64
Issue: 2
Year: 2014
Pages: 459-464
Summary lang: English
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Category: math
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Summary: 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 ${\rm nse}(G):=\{m_{k}\colon k \in \pi _{e}(G)\}$. 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 $G \cong L_{2}(p)$. (English)
Keyword: element order
Keyword: set of the numbers of elements of the same order
Keyword: linear group
MSC: 20D06
idZBL: Zbl 06391505
idMR: MR3277747
DOI: 10.1007/s10587-014-0112-y
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Date available: 2014-11-10T09:47:10Z
Last updated: 2016-07-01
Stable URL: http://hdl.handle.net/10338.dmlcz/144009
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