Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T09:48:19.905Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Brisley, Warren, ‘On varieties of metabelian p-groups and their laws’, J. Austral. Math. Soc. 7 (1967), 64–80.CrossRefGoogle Scholar
[2]Brisley, Warren, ‘Varieties of metabelian p-groups of class p, p+ 1’, J. Austral. Math. Soc. 12 (1971), 53–62.Google Scholar
[3]Fitzpatrick, Patrick, Varieties of nilpotent groups of class at most four (Ph.D. thesis, Australian National University, Canberra, 1980).Google Scholar
[4]Gupta, N. D. and Newman, M. F., ‘On metabelian groups’, J. Austral. Math. Soc. 6 (1966), 362–368.CrossRefGoogle Scholar
[5]Harris, L. F., Varieties and section closed classes of groups (Ph.D. thesis, Australian National University, Canberra, 1973).Google Scholar
[6]Heineken, Hermann, ‘Über ein Levisches Nilpotenzkriterium’, Arch. Math. (Basel) 12 (1961), 176–178.CrossRefGoogle Scholar
[7]Hermes, Hans, Einführung in die Verbandstheorie (Die Grundlehren der mathematischen Wissenschaften, 73. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1955).Google Scholar
[8]Huppert, B., Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).Google Scholar
[9]Jonsson, Bjarni, ‘Varieties of groups of nilpotency three’, Notice Amer. Math. Soc. 13 (1966), 488.Google Scholar
[10]Kljacko, A. A., ‘Varieties of p-groups of small class’, (Russian), Ordered Sets and Lattices No. 1, 31–42 (Izdat. Saratov. Univ., Saratov, 1971).Google Scholar
[11]Kovács, L. G., ‘Varieties of nilpotent groups of small class’, Topics in algebra, Proc. 18th SRI, [edited by Newman, M. F.] (Lecture Notes in Mathematics, 697, pp. 205–229. Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[12]Kovács, L. G., ‘The thirty-nine varieties’, The Math. Scientist 4 (1979), 113–128.Google Scholar
[13]Neumann, Hanna, Varieties of Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 37. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[14]Pentony, Paul, Laws in torsion free nilpotent varieties (Ph. D. thesis, Australian National University, Canberra, 1970).Google Scholar
See also: Abstract, Bull. Austral. Math. Soc. 5 (197), 283–284.Google Scholar
[15]Remeslennikov, V. N., ‘Two remarks on 3-step nilpotent groups’, (Russian), Algebra i Logika Sem. 4 (1965), no. 2, 59–65.Google Scholar
[16]Thrall, Robert M., ‘A note on a theorem by Witt’, Bull. Amer. Math. Soc. 47 (1941), 303–308.Google Scholar