On a Generalization of Baer Theorem
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.
Cite
@article{arxiv.1111.0224,
title = {On a Generalization of Baer Theorem},
author = {L. A. Kurdachenko and J. Otal and I. YA. Subbotin},
journal= {arXiv preprint arXiv:1111.0224},
year = {2011}
}