On one generalization of finite nilpotent groups

Abstract

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

16 pages