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


Finite groups whose subnormal subgroups permute with all Sylow subgroups

Author: Ram K. Agrawal
Journal: Proc. Amer. Math. Soc. 47 (1975), 77-83
MSC: Primary 20D35
MathSciNet review: 0364444
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Abstract: As a generalization of $ (t)$-groups and of $ (q)$-groups, a group $ G$ is called a $ (\pi - q)$-group if every subnormal subgroup of $ G$ permutes with all Sylow subgroups of $ G$. It is shown that if $ G$ is a finite solvable $ (\pi - q)$-group, then its hypercommutator subgroup $ D(G)$ is a Hall subgroup of odd order and every subgroup of $ D(G)$ is normal in $ G$; conversely, if a group $ G$ has a normal Hall subgroup $ N$ such that $ G/N$ is a $ (\pi - q)$-group and every subnormal subgroup of $ N$ is normal in $ G$, then $ G$ is a $ (\pi - q)$-group.

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PII: S 0002-9939(1975)0364444-4
Keywords: $ \pi $-quasinormal subgroup, $ (\pi - q)$-group
Article copyright: © Copyright 1975 American Mathematical Society