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Proceedings of the American Mathematical Society

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

 
 

 

On Sylow intersections in finite groups


Author: Geoffrey R. Robinson
Journal: Proc. Amer. Math. Soc. 90 (1984), 21-24
MSC: Primary 20D20
DOI: https://doi.org/10.1090/S0002-9939-1984-0722407-6
MathSciNet review: 722407
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Abstract: In general, for a given prime $p$ and finite group $G$, there need not be Sylow $p$-subgroups $P$ and $Q$ of $G$ with $P \cap Q = {O_p}(G)$. In this paper we show that if $G$ is $p$-soluble, and $p$ is not 2 or a Mersenne prime, then such Sylow $p$-subgroups exist (also we give conditions guaranteeing the existence of such Sylow subgroups when $p$ is 2 or a Mersenne prime). We also show that if $G$ is not $p$-soluble, but $p$ is odd and the components of $G/{O_p}(G)$ are in a certain class of quasi-simple groups, then there are Sylow $p$-subgroups $P$ and $Q$ of $G$ with $P \cap Q = {O_p}(G)$, unless perhaps $p$ is a Mersenne prime. When $G$ is $p$-soluble, our work extends results of N. Itô [2].


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Article copyright: © Copyright 1984 American Mathematical Society