On $\Pi $-property of some maximal subgroups of Sylow subgroups of finite groups.
(English).Czechoslovak Mathematical Journal,
vol. 73
(2023),
issue 4,
pp. 1349-1358
Summary: Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the $\Pi $-property in $G$ if for any chief factor $L / K$ of $G$, $| G / K : N_{G / K} ( HK/K\cap L/K )|$ is a $\pi (HK/K\cap L/K) $-number. We study the influence of some $p$-subgroups of $G$ satisfying the $\Pi $-property on the structure of $G$, and generalize some known results.