On one generalization of finite nilpotent groups

Abstract

Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set $\Bbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $σ$-central if the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is a $σ_{i}$-group for some $i=i(H/K)$. $G$ is called $σ$-nilpotent if every chief factor of $G$ is $σ$-central. We say that $G$ is semi-$σ$-nilpotent (respectively weakly semi-$σ$-nilpotent) if the normalizer $N_{G}(A)$ of every non-normal (respectively every non-subnormal) $σ$-nilpotent subgroup $A$ of $G$ is $σ$-nilpotent. In this paper we determine the structure of finite semi-$σ$-nilpotent and weakly semi-$σ$-nilpotent groups.