On the Normality of $\mathfrak F^{\omega}$-Abnormal Maximal Subgroups of Finite Groups

In the paper, the notion of an $\mathfrak F^{\omega}$-abnormal (and $\mathfrak F^{\omega}$-normal) maximal subgroup of a finite group is introduced, where $\mathfrak F$ is a nonempty class of groups and $\omega$ is a nonempty set of primes. The relationship between the $\mathfrak F^{\omega}$-abnormal maximal and normal subgroups is studied. Conditions are established under which $\mathfrak F^{\omega}$-abnormal maximal subgroups in a finite group are normal.