Criteria for $\pi$-separability of a finite group

Throughout this paper all groups are finite and $G$ always denotes a finite group. The group $G$ is said to be $\pi$-separable if every chief factor of $G$ is either a $\pi$-group or a $\pi'$-group. A subgroup $A$ of $G$ is said to be $\pi,\pi'$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_n=G$ such that either $A_{i-1}\trianglelefteq A_i$ or $A_i/(A_{i-1})_{A_i}$ is a $\pi$-separable group for all $i = 1, \dots, n$. In this paper we study the influence of $\pi,\pi'$-subnormal subgroups on the structure of the group.