Geodesic
On $\Pi$-property and $\Pi$-normality of subgroups of finite groups. II
Algebra i logika, Tome 54 (2015) no. 3, pp. 326-350

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Let $H$ be a subgroup of a group $G$. We say that $H$ satisfies the $\Pi$-property in $G$ if $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K)$-number for any chief factor $L/K$ of $G$. If there is a subnormal supplement $T$ of $H$ in $G$ such that $H\cap T\le I\le H$ for some subgroup $I$ satisfying the $\Pi$-property in $G$, then $H$ is said to be $\Pi$-normal in $G$. Using these properties that hold for some subgroups, we derive new $p$-nilpotency criteria for finite groups.
Keywords: finite group, $\Pi$-property, $\Pi$-normal subgroup, $p$-nilpotency. 
@article{AL_2015_54_3_a2,
     author = {B. Li and T. Foguel},
     title = {On $\Pi$-property and $\Pi$-normality of subgroups of finite {groups.~II}},
     journal = {Algebra i logika},
     pages = {326--350},
     publisher = {mathdoc},
     volume = {54},
     number = {3},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2015_54_3_a2/}
}
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B. Li; T. Foguel. On $\Pi$-property and $\Pi$-normality of subgroups of finite groups. II. Algebra i logika, Tome 54 (2015) no. 3, pp. 326-350. http://geodesic.mathdoc.fr/item/AL_2015_54_3_a2/