On the upper central series of infinite groups

Two relevant theorems by R. Baer and P. Hall show that a group is finite over a term with finite ordinal type of its upper central series if and only if it is finite-by-nilpotent. Extending these results, we prove here that if G is any group, the hypercentre factor group G/Z̄(G) is finite if and onl...

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Bibliographic Details
Published in:	Proceedings of the American Mathematical Society Vol. 139; no. 2; pp. 385 - 389
Main Authors:	DE FALCO, M., DE GIOVANNI, F., MUSELLA, C., SYSAK, Y. P.
Format:	Journal Article
Language:	English
Published:	Providence, RI American Mathematical Society 01-02-2011
Subjects:	Exact sciences and technology General mathematics General, history and biography Group theory Group theory and generalizations Infinite series Integers Mathematical theorems Mathematics Sciences and techniques of general use Series convergence hypercentre hypercentral group Upper central series Finite group
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Summary:	Two relevant theorems by R. Baer and P. Hall show that a group is finite over a term with finite ordinal type of its upper central series if and only if it is finite-by-nilpotent. Extending these results, we prove here that if G is any group, the hypercentre factor group G/Z̄(G) is finite if and only if G contains a finite normal subgroup N such that G/N is hypercentral (where the hypercentre Z̄(G) of G is defined as the last term of its upper central series).
ISSN:	0002-9939 1088-6826
DOI:	10.1090/S0002-9939-2010-10625-1