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Varieties of nilpotent groups of class four (I)

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Patrick Fitzpatrick  and
L. G. Kovács
Patrick Fitzpatrick
Affiliation:
Department of MathematicsInstitute of Advanced Studies Australian National UniversityCanberra, ACTAustralia
L. G. Kovács
Affiliation:
Department of MathematicsInstitute of Advanced Studies Australian National UniversityCanberra, ACTAustralia
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Abstract

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This is the first of three papers (the others by the first author alone) which determine all varieties of nilpotent groups of class (at most) four. The initial step is to reduce the problem to two cases: varieties whose free groups have no elements of order 2, and varieties whose free groups have no nontrivial elements of odd order. The varieties of the first kind form a distributive lattice with respect to order by inclusion (which is not a sublattice in the lattice of all group varieties). We give an embedding of this lattice in the direct product of six copies of the lattice which consist of 0 (as largest element) and the odd positive integers ordered by divisibility. The six integer parameters so associated with a variety directly match a (finite) defining set of laws for the variety. We also show that the varieties of the second kind do form a sublattice in the lattice of all varieties. That (nondistributive) sublattice will be treated, in a similarly conclusive manner, in the subsequent papers of this series.

MSC classification

Secondary: 20E10: Quasivarieties and varieties of groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 1 , August 1983 , pp. 59 - 73
Copyright
Copyright © Australian Mathematical Society 1983

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