Finite groups with certain nΦ-subgroups

Abstract

Only finite groups are considered. A subgroup $H$ of a group $G$ is called an $n\Phi$-subgroup if there exists a normal subgroup $K$ such that $G=HK$ and $H\cap K$ is contained in the Frattini subgroup of $H$. The structure of a finite group is obtained in the following cases: all normal subgroups are $n\Phi$-subgroups; the Frattini subgroup of the group is trivial and every $n\Phi$-subgroup is normal; every $2$-maximal subgroup is an $n\Phi$-subgroup; every $3$-maximal subgroup is an $n\Phi$-subgroup; for all primes $p$, every subgroup of order $p^2$ is an $n\Phi$-subgroup. For an arbitrary formation $\mathfrak F$, it is established that in the $\mathfrak F$-residual of the a group, every non-trivial $\mathfrak F$-subgroup is not an $n\Phi$-subgroup.