On the F\mathcal{F}^*-norm of a finite group

  • Quanfu Yan

    China Agricultural University, Beijing, China; and Kent State University, USA
  • Zhencai Shen

    China Agricultural University, Beijing, China
On the $\mathcal{F}^*$-norm of a finite group cover
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Abstract

Let GG be a finite group and F\mathcal{F} be a non-empty formation. We define the F\mathcal{F}^*-norm, denoted by NF(G)N_{\mathcal{F}}^{*}(G), to be intersection of the normalizers of the F\mathcal{F}-residuals of all FF-subgroups of GG, where F=NFF=\mathcal{N}\mathcal{F} is the class of all groups whose F\mathcal{F}-residuals are nilpotent. In this paper, we research the properties of NF(G)N_{\mathcal{F}}^{*}(G) and investigate the relationship between NF(G)N_{\mathcal{F}}^{*}(G) and NF(G),N_{\mathcal{F}}(G), where NF(G)N_{\mathcal{F}}(G) is the intersection of the normalizers of the F\mathcal{F}-residuals of all subgroups of G.G. We show that NF(G)=NF(G)N_{\mathcal{F}}^{*}(G)=N_{\mathcal{F}}(G) if AFN\mathcal{A}\subseteq \mathcal{F}\subseteq\mathcal{N}.

Cite this article

Quanfu Yan, Zhencai Shen, On the F\mathcal{F}^*-norm of a finite group. Rend. Sem. Mat. Univ. Padova 145 (2021), pp. 181–190

DOI 10.4171/RSMUP/77