On $\mathcal {F}$-abnormal maximal subgroups of a finite solvable group

Abstract:

Let $\Delta (G)$ be the intersection of the nonnormal maximal subgroup of a finite group. W. Gaschütz has shown that $\Delta (G)$ is nilpotent and that $\Delta (G)/\Phi (G)$ is the center of $G/\Phi (G)$. This note, by considering the intersection of the $\mathfrak {F}$-abnormal maximal subgroups, generalizes these results for a saturated formation $\mathfrak {F}$.