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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
DOI: https://doi.org/10.1090/S0002-9939-1975-0372022-6
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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Keywords: <IMG WIDTH="21" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$p’$">-Hall subgroup, <!– MATH $Z(J(P)),{O_p}(G)$ –> <IMG WIDTH="145" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$Z(J(P)),{O_p}(G)$">, <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img3.gif" ALT="$p$">-solvable, <!– MATH ${\operatorname {Syl} _p}(G)$ –> <IMG WIDTH="73" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img18.gif" ALT="${\operatorname {Syl} _p}(G)$">
Article copyright: © Copyright 1975 American Mathematical Society