$p$-子群具有$p$-超可解正规化子的有限群
Finite Groups with $p$-Supersolvable Normalizers of $p$-Subgroups

DOI：10.3770/j.issn:2095-2651.2022.05.004

 作者 单位 邱婷婷 盐城生物工程高等职业技术学校, 江苏 盐城 224051 吴金莲 西华师范大学数学与信息学院, 四川 南充 637009 张佳 西华师范大学数学与信息学院, 四川 南充 637009

在已有研究中,对于$p$-子群的正规化子而言,它的$p$-幂零性质对有限$p$-幂零群的结构具有重要的影响. 本文中, 设$P$是群$G$的西罗$p$-子群, $1\leq p^d<|P|$, 对于$P$的每个阶为$p^d$的正规子群$H$H,将$N_G(H)$的$p$-幂零性质减弱为$p$-超可解性质,结合$H$的弱$M$-可补充性质,探究$p$-超可解群的结构.同时,在$N_G(P)$是$p$-幂零的条件下,利用子群$K$的弱$M$-可补充条件研究群的$p$-幂零性质,其中$K_p\leq K$且$P'\leq K_p\leq \Phi(P)$. $K_p$是$K$的西罗$p$-子群.在一定程度上,主要结果推广了Frobenius定理.

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.