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

TL;DR

This paper generalizes Baer's theorems by characterizing formations of groups where intersections of certain subgroup normalizers equal the group's hypercenter, extending classical results to broader subgroup systems.

Contribution

It provides a complete description of formations where intersections of normalizers or subnormalizers of specific subgroup systems equal the hypercenter, generalizing Baer's theorems.

Findings

Characterized formations where normalizer intersections equal the hypercenter.

Described formations where subnormalizer intersections match the hypercenter.

Extended classical theorems to broader subgroup systems.

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 $G$.