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Varieties of nilpotent groups of class four (II)

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Patrick Fitzpatrick
Patrick Fitzpatrick
Affiliation:
Department of MathematicsUniversity College Cork, Ireland
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Abstract

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The first paper (written jointly with L. G. Kovács) of this three-part series reduced the problem of determining all varieties of the title to the study of the varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. The present paper deals with these under the additional assumption that the variety contains all nilpotent groups of class three. We label each such variety by a vector of eleven parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used, and matches directly a (finite) defining set of laws for the variety it labels. Moreover, one can readily recognize from the parameters whether one variety is contained in another. The third paper will complete the determination of all varieties of nilpotent groups of class four.

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. 74 - 108
Copyright
Copyright © Australian Mathematical Society 1983

References

Boerner, H. (1963), Representations of groups (North-Holland, Amsterdam).Google Scholar
Fitzpatrick, Patrick (1980), Varieties of nilpotent groups of class four (Ph. D. thesis, Australian National University, Canberra).CrossRefGoogle Scholar
Fitzpatrick, Patrick and Kovács, L. G. (1983), ‘Varieties of nilpotent groups of class four (I)’, J. Austral. Math. Soc. Ser. A 35, 59–73.CrossRefGoogle Scholar
Kovác, L. G. (1978), ‘Varieties of nilpotent groups of small class’, Topics in algebra (Proc. 18th SRI, Lecture Notes in Mathematics, 697, pp. 205–229. Springer-Verlag, Berlin, Heidelberg, New York).Google Scholar
Neumann, Hanna (1967), Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 37. Springer-Verlag, Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Wall, G. E. (1978), ‘Lie methods in group theory’, Topics in algebra (Proc. 18th SRI, Lecture Notes in Mathematics, 697, pp. 137–713. Springer-Verlag, Berlin, Heidelberg, New York).CrossRefGoogle Scholar