On a Generalization of Baer Theorem

TL;DR

This paper provides a simplified proof of a generalization of Baer's theorem, showing that if a group's quotient by its upper hypercenter is finite, then the group has a finite normal subgroup with a hypercentral quotient.

Contribution

The paper offers a new, shorter proof of a recent generalization of Baer's theorem and quantifies the size of the finite normal subgroup in terms of the quotient's order.

Findings

Provided a simpler proof of the generalization of Baer's theorem.

Established an explicit bound on the size of the finite normal subgroup.

Enhanced understanding of the structure of groups with finite quotients by their hypercenter.

Abstract

R. Baer has proved that if the factor-group G/{\zeta}_{n}(G) of a group G by the member {\zeta}_{n}(G) of its upper central series is finite (here n is a positive integer) then the member {\gamma}_{n+1}(G) of the lower central series of G is also finite. In particular, in this case, the nilpotent residual of G is finite. This theorem admits the following simple generalization that has been published very recently by M. de Falco, F. de Giovanni, C. Musella and Ya. P. Sysak: "If the factor-group G/Z of a group G modulo its upper hypercenter Z is finite then G has a finite normal subgroup L such that G/L is hypercentral". In the current article we offer a new simpler very short proof of this theorem and specify it substantially. In fact, we prove that if |G/Z|=t then |L|\leqt^{k}, where k=(1/2)(log_{p}t+1), and p is the least prime divisor of t.