Finite Groups with $\sigma$-Abnormal Schmidt Subgroups

Abstract

Let $G$ be a finite group and let $\sigma=\{\sigma_{i}\mid i\in I\}$ be a partition of the set of all primes $𝕇$. The group $G$ is $\sigma$-primary if $G$ is a $\sigma_{i}$-group for some $i\in I$; while $G$ is $\sigma$-nilpotent if $G$ is the direct product of $\sigma$-primary subgroups; and $G$ is a Schmidt group if $G$ is nonnilpotent but each proper subgroup in $G$ is nilpotent. A subgroup $A$ of $G$ is ${\sigma}$-abnormal in $G$ if for all subgroups $K<H$ in $G$, where $A\leq K$, the quotient group $H/K_{H}$ is not $\sigma$-primary. We describe the structure of finite groups whose every non-$\sigma$-nilpotent Schmidt subgroup is $\sigma$-abnormal.