# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## On Sylow intersections in finite groupsHTML articles powered by AMS MathViewer

by Geoffrey R. Robinson
Proc. Amer. Math. Soc. 90 (1984), 21-24 Request permission

## 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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