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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On relative normal complements in finite groups. III


Author: Henry S. Leonard
Journal: Proc. Amer. Math. Soc. 95 (1985), 5-6
MSC: Primary 20D40
MathSciNet review: 796435
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Abstract: Let $ G$ be a finite group, let $ H \leq G$, and let $ \pi $ be the set of prime divisors of $ \left\vert H \right\vert$. Assume that whenever two elements of $ H$ are $ G$-conjugate then they are $ H$-conjugate. Assume that for all $ h \in {H^\char93 }$, $ ({C_G}(h):{C_H}(h))$ is a $ \pi '$-number. We prove that $ H$ is a $ \pi $-Hall subgroup and that there exists a normal complement $ {G_0} = G - {({H^\char93 })^{G,\pi }}$. An example shows that the generalization to relative normal complements is not true.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0796435-X
PII: S 0002-9939(1985)0796435-X
Article copyright: © Copyright 1985 American Mathematical Society