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On Central Series

Published online by Cambridge University Press:  20 January 2009

I. D. Macdonald
I. D. Macdonald
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra
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Let

and

be, respectively, the upper and lower central series of a group G. Our purpose in this note is to extend known results and find some information as to which of the factors Zk/Zk−1 and Γk/Γk+1 may be infinite. Though our conclusions about the lower central series will be quite general we assume in the other case that the group is f.n., i.e. an extension of a finite group by a nilpotent group. The essential facts about f.n. groups are to be found in P. Hall's paper (4). We also refer to (4) for general notation; we reserve the letter k for positive integers.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 13 , Issue 2 , December 1962 , pp. 175 - 178
Copyright
Copyright © Edinburgh Mathematical Society 1962

References

(1) Baer, R., Representations of groups as quotient groups, Trans. Anier. Math. Soc., 58 (1945), 295–419.Google Scholar
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(3) Baer, R., Finite extensions: of abelian groups with minimum condition, Trans. Amer. Math. Soc., 79 (1955), 521–540.CrossRefGoogle Scholar
(4) Hall, P., Finite-by-nilpotent groups, Proc. Camb. Phil. Soc., 52 (1956), 611–616.CrossRefGoogle Scholar
(5) Macdonald, I. D., On certain varieties of groups, Math. Zeit., 76 (1961), 270–282.CrossRefGoogle Scholar