| Title:
|
A variation of Thompson's conjecture for the symmetric groups (English) |
| Author:
|
Abedei, Mahdi |
| Author:
|
Iranmanesh, Ali |
| Author:
|
Shirjian, Farrokh |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
743-755 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a finite group and let $N(G)$ denote the set of conjugacy class sizes of $G$. Thompson's conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N(G)=N(S)$, then $G\cong S$. In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G\cong {\rm Sym}(p+1)$ if and only if $|G|=(p+1)!$ and $G$ has a special conjugacy class of size $(p + 1)!/p$, where $p>5$ is a prime number. Consequently, if $G$ is a centerless group with $N(G)=N({\rm Sym}(p+1))$, then $G \cong {\rm Sym}(p+1)$. (English) |
| Keyword:
|
Thompson's conjecture |
| Keyword:
|
conjugacy class size |
| Keyword:
|
symmetric groups |
| Keyword:
|
prime graph |
| MSC:
|
20D08 |
| MSC:
|
20D60 |
| idZBL:
|
07250686 |
| idMR:
|
MR4151702 |
| DOI:
|
10.21136/CMJ.2020.0501-18 |
| . |
| Date available:
|
2020-09-07T09:37:30Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148325 |
| . |
| Reference:
|
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| . |
