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A Comment on Finite Nilpotent Groups of Deficiency Zero

Published online by Cambridge University Press:  20 November 2018

Edmund F. Robertson
Affiliation:
University of St. Andrews, Mathematical Institute, North Haugh, St. Andrews, Scotland
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Extract

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD 0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD 0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD 0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD 0-groups which contains nilpotent groups of arbitrarily large nilpotent class.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Beyl, F.R., The Schur multiplicator of metacyclic groups, Proc. Amer. Math. Soc. 40 (1973), 413-418.CrossRefGoogle Scholar
2. Campbell, C.M., Computational Techniques and the Structure of Groups in a Certain Class, in Proc. SYMSAC '76, 312-321. Association for Computing Machinery, New York, 1976.CrossRefGoogle Scholar
3. Macdonald, I.D., On a class of finitely presented groups, Canad. J. Math. 14 (1962), 602-613.CrossRefGoogle Scholar
4. Macdonald, I.D., A computer application for finite p-groups, J. Austral. Math. Soc. 17 (1974), 102-112.CrossRefGoogle Scholar
5. Mennicke, J., Einige endliche Gruppen mit drei Ergeugenden und drei Relationen, Arch. Math. (Basel) 10 (1959), 409-418.CrossRefGoogle Scholar
6. Wamsley, J.W., The deficiency of metacyclic groups, Proc. Amer. Math. Soc. 24 (1970) 724-726.Google Scholar
7. Wamsley, J.W., A class of three-generator, three relation, finite groups, Canad. J. Math. 22 (1970), 36-40.CrossRefGoogle Scholar
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