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On varieties of metabelian groups of prime-power exponent

Published online by Cambridge University Press:  09 April 2009

M. S. Brooks
M. S. Brooks
Affiliation:
Australian National University
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Let Un denote the variety of abelian groups of exponent dividing n, and let p be an arbitrary prime. In this paper all non-nilpotent, join-ireducible subvarieties of the product variety UpUp2 are determined. The proper subvarieties of this kind in fact form an infinite ascending chain …, and an arbitrary proper subvariety B of UpUp2 is either nilpotent or a join , where L is nilpotent and k is uniquely determined by B.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 2 , September 1972 , pp. 129 - 154
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Brisely, Warren, ‘On varieties of metabelian p-groups and their laws’, J. Austral. Math. Soc. 7 (1967), 64–80.CrossRefGoogle Scholar
[2]Brooks, M. S., ‘On lattices of varieties of metabelian groups’, J. Austral. Math. Soc. 12 (1971), 161–166.CrossRefGoogle Scholar
[3]Bryce, R. A., ‘Metabelian groups and varieties’, Philos. Trans. Roy. Soc. London, Ser. A. 266 (1970) 281–355.Google Scholar
[4]Cohen, D. E., ‘On the laws of a metabelian variety’, J. Algebra 5 (1967), 267–273.CrossRefGoogle Scholar
[5]Kovács, L. G., ‘The descending chain condition in join-continuous modular lattices’, J. Austral. Math. Soc. 10 (1969), 1–4.CrossRefGoogle Scholar
[6]Kov´cs, L. G. and Newman, M. F., ‘On varieties of metabelian groups’, in preparation.Google Scholar
[7]Kovács, L. G. and Newman, M. F., ‘On non-Cross varieties of groups’, J. Austral. Math. Soc. 12 (1971), 129–144.CrossRefGoogle Scholar
[8]Liebeck, Hans, ‘Concerning nilpotent wreath products’, Proc. Camb. Philos. Soc. 58 (1962), 443–451.CrossRefGoogle Scholar
[9]Lyndon, R. C., ‘Two notes on nilpotent groups’, Proc. Amer. Math. Soc. 3 (1952), 579–583.Google Scholar
[10]Meier-Wunderli, H., ‘Metabelsche Gruppen’, Comment. Math. Helvet. 25 (1951), 1–10.CrossRefGoogle Scholar
[11]Neumann, Hanna, Varieties of groups (Springer-Verlag, 1967).CrossRefGoogle Scholar
[12]Newman, T. G., ‘The descending chain condition in modular lattices’, J. Austral. Math. Soc. (to appear).Google Scholar
[13]Paul, M. Weichsel, ‘On metabelianp-groups’, J. Austral. Math. Soc. 7 (1967), 55–63.Google Scholar