## On $$p$$-nilpotency of finite groups with some subgroups $$\pi$$-quasinormally embedded.(English)Zbl 1094.20007

A subgroup $$X$$ of a finite group $$G$$ is called $$\pi$$-quasinormal if $$XP=PX$$ for each Sylow subgroup $$P$$ of $$G$$. Moreover, a subgroup $$H$$ of $$G$$ is said to be $$\pi$$-quasinormally embedded in $$G$$ if for each prime divisor $$p$$ of $$|H|$$ any Sylow $$p$$-subgroup of $$H$$ is also a Sylow $$p$$-subgroup of some $$\pi$$-quasinormal subgroup of $$G$$. It has been proved by M. Asaad and A. A. Heliel [J. Pure Appl. Algebra 165, No. 2, 129-135 (2001; Zbl 1011.20019)] that if $$p$$ is the smallest prime divisor of the order of a finite group $$G$$, then $$G$$ is $$p$$-nilpotent if and only if all maximal subgroups of Sylow $$p$$-subgroups of $$G$$ are $$\pi$$-quasinormally embedded in $$G$$.
In the paper under review the authors obtain some $$p$$-nilpotency criteria for finite groups related to the assumption that certain special subgroups are $$\pi$$-quasinormally embedded. For instance, they prove that if a finite group $$G$$ contains a Sylow $$p$$-subgroup $$P$$ such that $$N_G(P)$$ is $$p$$-nilpotent and all maximal subgroups of $$P$$ are $$\pi$$-quasinormally embedded in $$G$$, then $$G$$ itself is $$p$$-nilpotent.

### MSC:

 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D40 Products of subgroups of abstract finite groups 20D15 Finite nilpotent groups, $$p$$-groups

Zbl 1011.20019
Full Text: