What is the Glauberman-Thompson theorem? I read in a paper, if $% N_{N}(Z(J(P))$ is nilpotent (where $P$ is a Sylow $p$-subgroup of $N$ and $N$ is a minimal normal subgroup of $G$), then by the Glauberman-Thompson theorem, $N$ is $p$-nilpotent.
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The Glauberman-Thompson Theorem probably refers to the following Theorem: Let $p$ be an odd prime, and $G$ is a finite group with Sylow p-subgroup $P$. Then if $N_{G}(ZJ(P))$ has a normal $p$-complement, so does $G.$ Here, $J(P)$ is the subgroup of $P$ generated by the Abelian subgroups of $P$ of maximal order, and $ZJ(P)$ is its center. The proof is difficult, but can be found in many modern Group Theory texts. The result was a culmination of a series of results: Thompson first proved that if $C_{G}(Z(P))$ and $N_{G}(J(P))$ both have normal $p$-complements, so does $G$ (when $p$ is odd).
Geoff Robinson
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