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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On two theorems of Thompson


Author: Guang Xiang Zhang
Journal: Proc. Amer. Math. Soc. 98 (1986), 579-582
MSC: Primary 20D20
DOI: https://doi.org/10.1090/S0002-9939-1986-0861754-6
MathSciNet review: 861754
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Abstract: Let $ G$ be a finite group.

Theorem. Let $ P \in {\operatorname{Syl} _p}(G)$ with $ {\Omega _1}(P) \leq Z(P)$. If $ {N_G}(Z(P))$ has a normal $ p$-complement, then so does $ G$.

Corollary. Let $ M$ be a nilpotent maximal subgroup of $ G$ and $ P \in {\operatorname{Syl} _2}(M)$ with $ {\Omega _2}(P) \leq Z(P)$. Then $ G$ is solvable.

This extends Thompson's solvability theorem [9]. We also give two other results generalizing Thompson's theorem.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0861754-6
Article copyright: © Copyright 1986 American Mathematical Society