On generalizations of Baer's theorems about the hypercenter of a finite group

Abstract: We investigate the intersection of normalizers and $\mathfrak{F}$-subnormalizers of different types of systems of subgroups ($\mathfrak{F}$-maximal, Sylow, cyclic primary). We described all formations $\mathfrak{F}=\underset{i\in I}\times\mathfrak{F}_{\pi_i}$ for which the intersection of normalizers of all $\mathfrak{F}_i$-maximal subgroups of $G$ is the $\mathfrak{F}$-hypercenter of $G$ for every group $G$. Also we described all formations $\mathfrak{F}$ for which the intersection of $\mathfrak{F}$-subnormalizers of all Sylow (cyclic primary) subgroups of $G$ is the $\mathfrak{F}$-hypercenter of $G$ for every group $G$.