Finite groups with given Schmidt subgroups

Throughout the article, all groups are finite and $G$ always denotes a finite group. A subgroup $H$ of the group $G$ is called $\mathfrak{U}_p$-normal in $G$ ($p$ is a prime) if every chief factor of the group $G$ between $H^G$ and $H_G$ is either cyclic or a $p'$-group. In this article, we prove that if each Schmidt subgroup of the group $G$ is either subnormal or $\mathfrak{U}_p$-normal in $G$, then the derived subgroup $G'$ of $G$ is $p$-nilpotent. Some well-known results are generalized.