Variants of theorems of Baer and Hall on finite-by-hypercentral groups

TL;DR

This paper extends classical theorems by Baer and Hall to finite-by-hypercentral groups, establishing bounds on the hypercenter index based on a finite normal subgroup, with applications to automorphism groups.

Contribution

It generalizes Baer and Hall's theorems to finite-by-hypercentral groups and provides bounds on the hypercenter index related to finite normal subgroups.

Findings

Bound on the hypercenter index in finite-by-hypercentral groups

Completion of recent generalizations of classical theorems

Applications to automorphism groups acting on ascending series

Abstract

We show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical theorems by R. Baer and P. Hall. Then we apply our results to groups of automorphisms of a group $G$ acting in a restricted way on an ascending normal series of $G$.