A characterization of the linear groups $L_{2}(p)$

Title:

A characterization of the linear groups $L_{2}(p)$

Creator:

Khalili Asboei, Alireza and Iranmanesh, Ali

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:ee9b3372-43cf-44d6-89ac-40f9d3a70ed5
uuid:ee9b3372-43cf-44d6-89ac-40f9d3a70ed5

Subject:

element order, set of the numbers of elements of the same order, and linear group

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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 $G \cong L_{2}(p)$.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

459-464

Source:

Czechoslovak Mathematical Journal | 2014 Volume:64 | Number:2

Harvested from:

CDK

Metadata only:

false