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On varieties of metabelian p-groups, and their laws

Published online by Cambridge University Press:  09 April 2009

Warren Brisley
Warren Brisley
Affiliation:
Department of Mathematics University of Newcastle
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The purpose of this article is to present some results on varieties of metabelian p-groups, nilpotent of class c, with the prime p greater than c. After some preliminary lemmas in § 3, it is established in § 4, Theorem 3, that there is a simple basis for the laws of such a variety, and this basis is explicitly stated. This allows the description of the lattice of such varieties, and in § 5, Theorem 4, it is shown that each such variety has a two-generator member which generates it; this is established by the help of Theorem 5, which states that each critical group is a two-generator group, and Theorem 6, which gives explicitly the varieties generated by the proper subgroups, by the proper quotient groups, and by the proper factor groups of such a critical group.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 1 , February 1967 , pp. 64 - 80
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Various Sources; e.g. Abramowitz, M. and Stegun, I. A., ‘Handbook of Mathematical Functions’ (National Bureau of Standards, 1964).Google Scholar
[2]Baumslag, G., Neumann, B. H., Neumann, Hanna and Neumann, Peter M., ‘On varieties generated by a finitely generated group’, Math. Z. 86 (1964), 93–122.CrossRefGoogle Scholar
[3]Brisley, W., ‘A Problem of D. W. Barnes’, Proc. Internat. Conf. Theory of Groups, A. N. U., Canberra, 1965. (Gordon and Breach, New York, 1966.)Google Scholar
[4]Gupta, N. D. and Newman, M. F., ‘On metabelian groups’, J. Aust. Math. Soc., 6 (1966), 362–368.CrossRefGoogle Scholar
[5]Higman, Graham, ‘Some remarks on varieties of groups’, Quarterly J. Math. Oxford (2) 10 (1959), 165–178.CrossRefGoogle Scholar
[6]Kovács, L. G. and Newman, M. F., ‘Minimal Verbal Subgroups’, Proc. Cambridge Philos. Soc.Google Scholar
[7]Macdonald, I. D., ‘A theorem about critical p-groups’, Proc. Internat. Conf. Theory of Groups, A. N. U., Canberra, 1965, (Gordon and Breach, New York, 1966).Google Scholar
[8]Neumann, Hanna, Varieties of Groups (Duplicated lecture notes, Manchester College of Science and Technology, 1963).Google Scholar
[9]Neumann, Hanna (Book to be published by Springer).Google Scholar
[10]Weichsel, P. M., ‘On critical p-groups’, Proc. Lond. Math. Soc., 3rd series, 14 (1964), 83–100.CrossRefGoogle Scholar