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arXiv · 2402.13365

Some results on the norm of finite groups

Mark L. Lewis·Zhencai Shen·Quanfu Yan
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Abstract

Let $G$ be a finite group and $N_Ω(G)$ be the intersection of the normalizers of all subgroups belonging to the set $Ω(G),$ where $Ω(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this paper, we show that $N_Ω(G)= Z_{\infty}(G)$ if $Ω(G)$ is one of the following: (i) the set of all self-normalizing subgroups of $G$; (ii) the set of all subgroups of $G$ satisfying the subnormalizer condition in $G$; (iii) the set of all pronormal subgroups of $G$; (iv) the set of all $\mathscr{H}$-subgroups of $G$; (v) the set of all weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$.

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Mark L. Lewis, Zhencai Shen, Quanfu Yan. 2024-02-20. Some results on the norm of finite groups. https://arxiv.org/abs/2402.13365

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