Ricerche di Matematica, Volume (10.100711587), No (1), Year (2013-12) , Pages (1-5)

Title : ( A converse of Baer’s theorem )

Authors: Rasoul Hatamian , Mitra Hassanzadeh , Saeed Kayvanfar ,

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Schur’s classical theorem states that for a group G, if G/Z(G) is finite, then G is finite. Baer extended this theorem for the factor group G/Zn(G), inwhich Zn(G) is the n-th term of the upper central series of G. Hekster proved a converse of Baer’s theorem as follows: If G is a finitely generated group such that γn+1(G) is finite, then G/Zn(G) is finite where γn+1(G) denotes the (n+1)st term of the lower central series of G. In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that G/Zn(G) is finitely generated. Furthermore, we show that the index of the subgroup Zn(G) is bounded by a precisely determined function of the order of γn+1(G). Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that Z2n(G)/Zn(G) is finitely generated. Although in this case the bound for the order of γn+1(G) is not achieved.


, Baer’s theorem · n, Isoclinism of groups · Nilpotent groups
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author = {Hatamian, Rasoul and Hassanzadeh, Mitra and Kayvanfar, Saeed},
title = {A converse of Baer’s theorem},
journal = {Ricerche di Matematica},
year = {2013},
volume = {10.100711587},
number = {1},
month = {December},
issn = {0035-5038},
pages = {1--5},
numpages = {4},
keywords = {Baer’s theorem · n-Isoclinism of groups · Nilpotent groups},


%0 Journal Article
%T A converse of Baer’s theorem
%A Hatamian, Rasoul
%A Hassanzadeh, Mitra
%A Kayvanfar, Saeed
%J Ricerche di Matematica
%@ 0035-5038
%D 2013