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Proceedings of the American Mathematical Society

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



On normal complements of $ \mathfrak{F}$-covering subgroups

Author: H. J. Schmidt
Journal: Proc. Amer. Math. Soc. 25 (1970), 457-459
MSC: Primary 20.40
MathSciNet review: 0258960
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Abstract: If $ \mathcal{F}$ is a suitably restricted formation, we show that an $ \mathcal{F}$-covering subgroup $ H$ which is a Hall subgroup of the finite, solvable group $ G$ is complemented by the $ \mathcal{F}$-residual of $ G$, provided $ H$ normalizes an $ \mathcal{F}$-normalizer of $ G$. In particular, $ H$ is complemented by the $ \mathcal{F}$-residual, if $ H$ is an $ \mathcal{F}$-normalizer of $ G$. Further, if $ \mathcal{F}$ is the class of nilpotent groups, then $ H$ complements the nilpotent residual, if $ G$ has pronormal system normalizers. Examples are given to show the necessity of the various hypotheses.

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Keywords: Formation, solvable group, $ \mathcal{F}$-covering subgroup, $ \mathcal{F}$-normalizer, $ \mathcal{F}$-residual, Carter subgroup, Hall subgroup, pronormal subgroup, system normalizer
Article copyright: © Copyright 1970 American Mathematical Society

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