On a generalization of a theorem of Burnside

Shi, Jiangtao. "On a generalization of a theorem of Burnside." Czechoslovak Mathematical Journal 65.3 (2015): 587-591. <http://eudml.org/doc/271830>.

@article{Shi2015,
abstract = {A theorem of Burnside asserts that a finite group $G$ is $p$-nilpotent if for some prime $p$ a Sylow $p$-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 $p$-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 $n\ge 3$ for $p>2$ and $n\ge 4$ for $p=2$, then $G$ is $p$-nilpotent or $p$-closed.},
author = {Shi, Jiangtao},
journal = {Czechoslovak Mathematical Journal},
keywords = {non-cyclic $p$-subgroup; $p$-nilpotent; self-normalizing subgroup; normal subgroup},
language = {eng},
number = {3},
pages = {587-591},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a generalization of a theorem of Burnside},
url = {http://eudml.org/doc/271830},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Shi, Jiangtao
TI - On a generalization of a theorem of Burnside
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 3
SP - 587
EP - 591
AB - A theorem of Burnside asserts that a finite group $G$ is $p$-nilpotent if for some prime $p$ a Sylow $p$-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 $p$-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 $n\ge 3$ for $p>2$ and $n\ge 4$ for $p=2$, then $G$ is $p$-nilpotent or $p$-closed.
LA - eng
KW - non-cyclic $p$-subgroup; $p$-nilpotent; self-normalizing subgroup; normal subgroup
UR - http://eudml.org/doc/271830
ER -