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Proceedings of the American Mathematical Society

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

 

 

Hall subgroups and $ p$-solvability


Authors: A. Gonçalves and C. Y. Ho
Journal: Proc. Amer. Math. Soc. 52 (1975), 97-98
MSC: Primary 20D10
MathSciNet review: 0372022
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Abstract: Let $ G$ be a finite group and let $ \pi (G) = \{ p,{q_1}, \ldots ,{q_r}\} $ be the set of all prime divisors of $ G$. Suppose there is a $ p'$-Hall subgroup $ H$. If there are subgroups $ P,{Q_1}, \ldots ,{Q_r}$ such that $ P\epsilon {\operatorname{Syl} _p}(G),{Q_i}\epsilon {\operatorname{Syl} _{{q_i}}}(H)$, and $ {L_i} = P{Q_i}$ is a subgroup, $ i = 1, \ldots ,r$, then $ G$ is $ p$-solvable. Moreover, if the subgroup $ H$ is solvable, then $ G$ is solvable too.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0372022-6
Keywords: $ p'$-Hall subgroup, $ Z(J(P)),{O_p}(G)$, $ p$-solvable, $ {\operatorname{Syl} _p}(G)$
Article copyright: © Copyright 1975 American Mathematical Society