# 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 $G\cong {L}_{2}\left(p\right)$.

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