Finite Groups with $p$-Supersolvable Normalizers of $p$-Subgroups

Abstract:

      In the literature, $p$-nilpotency of the normalizers of $p$-subgroups has an important influence on finite $p$-nilpotent groups. In this paper, we extend the $p$-nilpotency to $p$-supersolvability and choose every normal $p$-subgroups $H$ of $P$ such that $|H|=p^{d}$ and explore $p$-supersolvability of $G$ by the conditions of weakly $\mathcal{M}$-supplemented properties of $H$ and $p$-supersolvability of the normalizer $N_{G}(H)$, where $1\leq p^{d}<|P|$. Also, we study the $p$-nilpotency of $G$ under the assumptions that $N_{G}(P)$ is $p$-nilpotent and the weakly $\cal M$-supplemented condition on a subgroup $K$ such that $K_{p}\unlhd K$ and $P'\leq K_{p} \leq\Phi(P)$, $K_{p}$ is a Sylow $p$-subgroup $K$. To some extent, our main results can be regarded as generalizations of the Frobenius theorem.