Author:
Fitzpatrick Patrick,Kovács L. G.
Abstract
AbstractThis 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.
Publisher
Cambridge University Press (CUP)
Subject
General Mathematics,Statistics and Probability
Cited by
4 articles.
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1. L. G. KOVÁCS AND VARIETIES OF GROUPS;Journal of the Australian Mathematical Society;2015-05-13
2. Identities;Encyclopaedia of Mathematical Sciences;1991
3. Varieties of nilpotent groups of class four (III);Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics;1983-08
4. Varieties of nilpotent groups of class four (II);Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics;1983-08