| Title:
|
On a generalization of a theorem of Burnside (English) |
| Author:
|
Shi, Jiangtao |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
3 |
| Year:
|
2015 |
| Pages:
|
587-591 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A theorem of Burnside asserts that a finite group $G$ is \mbox {$p$-nilpotent} if for some prime $p$ a Sylow \mbox {$p$-subgroup} of $G$ lies in the center of its normalizer. In this paper, let $G$ be a finite group and $p$ the smallest prime divisor of $|G|$, the order of $G$. Let $P\in {\rm Syl}_p(G)$. As a generalization of Burnside's theorem, it is shown that if every non-cyclic \mbox {$p$-subgroup} of $G$ is self-normalizing or normal in $G$ then $G$ is solvable. In particular, if $P\ncong \langle a,b\vert a^{p^{n-1}}=1,b^2=1, b^{-1}ab=a^{1+{p^{n-2}}}\rangle $, where $n\geq 3$ for $p>2$ and $n\geq 4$ for $p=2$, then $G$ is \mbox {$p$-nilpotent} or \mbox {$p$-closed}. (English) |
| Keyword:
|
non-cyclic $p$-subgroup |
| Keyword:
|
$p$-nilpotent |
| Keyword:
|
self-normalizing subgroup |
| Keyword:
|
normal subgroup |
| MSC:
|
20D10 |
| MSC:
|
20D20 |
| idZBL:
|
Zbl 06537682 |
| idMR:
|
MR3407595 |
| DOI:
|
10.1007/s10587-015-0198-x |
| . |
| Date available:
|
2015-10-04T17:59:25Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144433 |
| . |
| Reference:
|
[1] Huppert, B.: Endliche Gruppen I.Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134 Springer, Berlin German (1967). Zbl 0217.07201, MR 0224703 |
| Reference:
|
[2] Robinson, D. J. S.: A Course in the Theory of Groups.Graduate Texts in Mathematics 80 Springer, New York (1996). MR 1357169 |
| . |
