On a generalization of Baer Theorem

Abstract

R. Baer has proved that if the factor-group G/ζ n (G) of a group G by the member ζ n (G) of its upper central series is finite (here n is a positive integer) then the member γ 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| ≤ t k , where k = 1 2 (logpt + 1), and p is the least prime divisor of t.

On the structure of some infinite dimensional linear groups

“…This paper is concerned with obtaining a linear version of the main theorem of the papers [3,12]. To describe our work, we need some more terminology and notation.…”

“…It was proved in [12] that if G is a group, Z is the upper hypercenter of G and if G/Z is nite of order t, then G has a nite normal subgroup L, of order bounded in terms of t, such that G/L is hypercentral. A nonquantitative version of this result had earlier appeared in [3].…”

Linear Groups with Almost Right Engel Elements

“…In the present section we will establish two useful results about almost right Engel elements in infinite groups (see A well-known theorem of Baer states that if, for a group G and a positive integer k, the quotient G/Z k (G) is finite, then so is γ k+1 (G) (see [11, 14.5.1]). Here γ k+1 (G) denotes the term of the lower central series of G. Recently, the following related result was obtained in [1] (see also [9]). Proof.…”

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

“…Recall that the hypercenter of a group G is the last term of the upper central series of G (see details below). Then in [5] it has been shown that d may be bounded by a function of t, namely t (1+log 2 t)/2 . Here we complete the picture by showing that t in turn may be bounded by a function of d. There are many generalization and variants of Baer and Hall theorems.…”