On one generalization of finite nilpotent groups

TL;DR

This paper generalizes finite nilpotent groups by introducing $\sigma$-nilpotency based on prime partitions, and characterizes the structure of semi-$\sigma$-nilpotent and weakly semi-$\sigma$-nilpotent groups.

Contribution

It defines new classes of groups, $\sigma$-nilpotent, semi-$\sigma$-nilpotent, and weakly semi-$\sigma$-nilpotent, and determines their structural properties.

Findings

Characterization of $\sigma$-nilpotent groups.

Structural description of semi-$\sigma$-nilpotent groups.

Structural description of weakly semi-$\sigma$-nilpotent groups.

Abstract

Let $\sigma =\{\sigma_{i} | i\in I\}$ 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 $\sigma$-central if the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ 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.