On $\sigma$-subnormal subgroups and products of finite groups

Suppose that $\sigma =\{{\sigma }_i:i\in I\}$ is a partition of the set $\mathbb{P}$ of all primes. A subgroup $A$ of a finite group $G$ is said to be $\sigma$-subnormal in $G$ if $A$ can be joined to $G$ by a chain of subgroups $A=A_0\subseteq A_1\subseteq \cdots \subseteq A_n=G$ such that either $A_{i-1}$ normal in $A_i$ or $A_i/Cor{e}_{A_i}(A_{i-1})$ is a ${\sigma }_j$-group for some $j\in I$, for every $1\le i\le n$. A $\sigma$-subnormality criterion related to products of subgroups of finite $\sigma$-soluble groups is proved in the paper. As a consequence, a characterisation of the $\sigma$-Fitting subgroup of a finite $\sigma$-soluble group naturally emerges.