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

Let $G$ be a finite group and $\sigma =\{{\sigma }_i\mathrm{\mid }i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $$\mathbb{P}$=\bigcup_{i\in I}\sigma_{i}$and$\sigma_{i}\cap \sigma_{j}=\emptyset$forall$i\neq j$. A chief factor $H\mathrm{/}K$ of $G$ is said to be $\sigma$-central in $G$, if the semidirect product $(H\mathrm{/}K)⋊(G\mathrm{/}{C}_G(H\mathrm{/}K))$ 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.