On Sylow intersections in finite groups

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].