A generalization of Hall's theorem on hypercenter

TL;DR

This paper characterizes formations where the hypercenter and certain subgroup intersections coincide, generalizes classical results, and analyzes the connectivity of non-$\sigma$-nilpotent graphs.

Contribution

It describes all formations with a specific hypercenter property, solving a case of Shemetkov's problem and extending classical results by Hall and Baer.

Findings

Characterization of formations with hypercenter intersection property

Solution to a case of Shemetkov's problem

Non-$\sigma$-nilpotent graph is connected with diameter at most 3

Abstract

Let $ \sigma$ be a partition of the set of all primes and $\mathfrak{F}$ be a hereditary formation. We described all formations $\mathfrak{F}$ for which the $\mathfrak{F}$-hypercenter and the intersection of weak $K$-$\mathfrak{F}$-subnormalizers of all Sylow subgroups coincide in every group. In particular the formation of all $\sigma$-nilpotent groups has this property. With the help of our results we solve a particular case of L.A.~Shemetkov's problem about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter. As corollaries we obtained P. Hall's and R. Baer's classical results about the hypercenter. We proved that the non-$\sigma$-nilpotent graph of a group is connected and its diameter is at most 3.