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- Title
Varieties of nilpotent groups of class four (I).
- Authors
Fitzpatrick, Patrick; Kovács, L. G.
- Abstract
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.
- Publication
Journal of the Australian Mathematical Society, 1983, Vol 35, Issue 1, p59
- ISSN
1446-7887
- Publication type
Article
- DOI
10.1017/S1446788700024770