On the $\mathfrak{F}$-Norm of a Finite Group

Let $G$ be a finite group, and let $\mathfrak{F}$ be a nonempty formation. Then the intersection of the normalizers of the $\mathfrak{F}$-residuals of all subgroups of $G$ is called the $\mathfrak{F}$-norm of $G$ and is denoted by $N_{\mathfrak{F}}(G)$. A group $G$ is called $\mathfrak{F}$-critical if $G \not\in \mathfrak{F}$, but $U\in \mathfrak{F}$ for any proper subgroup $U$ of $G$. We say that a finite group $G$ is generalized $\mathfrak{F}$-critical if $G$ contains a normal subgroup $N$ such that $N\le \Phi (G)$ and the quotient group $G/N$ is $\mathfrak{F}$-critical. In this publication, we prove the following result: If $G$ does not belong to the nonempty hereditary formation $\mathfrak{F},$ then the $\mathfrak{F}$-norm $N_{\mathfrak{F}}(G)$ of $G$ coincides with the intersection of the normalizers of the $\mathfrak{F}$-residuals of all generalized $\mathfrak{F}$-critical subgroups of $G$. In particular$,$ the norm $N (G)$ of $G$ coincides with the intersection of the normalizers of all cyclic subgroups of $G$ of prime power order.