Finite Groups with $p$-Supersolvable Normalizers of $p$-Subgroups
Received:November 09, 2021  Revised:May 07, 2022
Key Words: normalizer   weakly $\cal M$-supplemented subgroup   $p$-supersolvability   $p$-nilpotency  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12001436), the Natural Science Foundation of Sichuan Province (Grant No.2022NSFSC1843) and Chunhui Plan Cooperative Scientific Research Project of Ministry of Education of the People's Republic of China and the Fundamental Research Funds of China West Normal University (Grant Nos.17E091; 18B032).
Author NameAffiliation
Tingting QIU Yancheng Biological Engineering Higher Vocational Technology School, Jiangsu 224051, P. R. China 
Jinlian WU School of Mathematics and Information, China West Normal University, Sichuan 637009, P. R. China 
Jia ZHANG School of Mathematics and Information, China West Normal University, Sichuan 637009, P. R. China 
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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.
Citation:
DOI:10.3770/j.issn:2095-2651.2022.05.004
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