Geodesic
A characterization of the linear groups $L_{2}(p)$
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 459-464
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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 $G \cong L_{2}(p)$.
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 $G \cong L_{2}(p)$.
DOI : 10.1007/s10587-014-0112-y
Classification : 20D06
Keywords: element order; set of the numbers of elements of the same order; linear group 
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Khalili Asboei, Alireza; Iranmanesh, Ali. A characterization of the linear groups $L_{2}(p)$. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 459-464. doi: 10.1007/s10587-014-0112-y

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