On the \(\mathcal{F}^*\)-norm of a finite group (Q2039376)

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