| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2014-11-10T09:47:10Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144009 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| . |
