On finite groups factorized by $\sigma$-nilpotent subgroups

TL;DR

This paper investigates the structure of finite groups that can be expressed as the product of two $\sigma$-nilpotent subgroups, extending existing results in the theory of group factorizations and $\sigma$-nilpotency.

Contribution

It generalizes known results by analyzing finite groups factorized by two $\sigma$-nilpotent subgroups, providing new insights into their properties.

Findings

Finite groups factorized by two $\sigma$-nilpotent subgroups have specific structural properties.

The results extend classical theorems on group factorizations to the $\sigma$-nilpotent context.

New criteria for $\sigma$-nilpotency in product groups are established.

Abstract

Let $G$ be a finite group and $\sigma=\{\sigma_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}\sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}=\emptyset$ for all $i\neq j$. A chief factor $H/K$ of $G$ is said to be $\sigma$-central in $G$, if the semidirect product $(H/K)\rtimes(G/C_G(H/K))$ is a $\sigma_i$-group for some $i\in I$. The group $G$ is said to be $\sigma$-nilpotent if either $G=1$ or every chief factor of $G$ is $\sigma$-central. In this paper, we study the properties of a finite group $G=AB$, factorized by two $\sigma$-nilpotent subgroups $A$ and $B$, and also generalize some known results.