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non-cyclic $p$-subgroup; $p$-nilpotent; self-normalizing subgroup; normal subgroup
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}.
[1] Huppert, B.: Endliche Gruppen I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134 Springer, Berlin German (1967). MR 0224703 | Zbl 0217.07201
[2] Robinson, D. J. S.: A Course in the Theory of Groups. Graduate Texts in Mathematics 80 Springer, New York (1996). MR 1357169
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