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Some conditions for finite presentation of nilpotent-by-abelian groups

Published online by Cambridge University Press:  24 October 2008

A. J. McIsaac
A. J. McIsaac
Affiliation:
Dulwich College, London SE 21
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Extract

Bieri and Strebel [2] have established a striking criterion that determines whether a metabelian group is finitely presented. This criterion is of a sort that implies that all homomorphic images of a finitely presented metabelian group are finitely presented. Another of their results is that the class of nilpotent-of-class-two-by-abelian groups has the property that all homomorphic images of finitely presented groups in this class are finitely presented. However, they point out that an example due to Abels [1] shows that the class of nilpotent-of-class-three-by-abelian groups does not have this property. One could not therefore expect that a criterion of the same type as that of Bieri and Strebel would distinguish the finitely presented groups in general classes of nilpotent-by-abelian groups. One might hope, however, to find such a criterion for nilpotent-of-class-two-by-abelian groups. The aim of this paper is to make some steps in this direction, by finding a sufficient condition for such groups to be finitely presented.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 3 , May 1984 , pp. 411 - 415
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Abels, H.. An example of a finitely presented soluble group. In Homological Group Theory, London Mathematical Society Lecture Notes Series 36 (Cambridge University Press, 1979), pp. 205–211.CrossRefGoogle Scholar
[2]Bieri, R. and Strebel, R.. Valuations and finitely presented metabelian groups. Proc. London Math. Soc. (3) 41 (1980), 439–464.CrossRefGoogle Scholar
[3]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers (Oxford University Press, 1938).Google Scholar