Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T12:08:53.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Preservation of saturation and stability in a variety of nilpotent groups

Published online by Cambridge University Press:  12 March 2014

Pat Rogers
Pat Rogers*
Affiliation:
York University, Toronto, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

This paper is a contribution to the growing literature on the model theory of nilpotent groups. (See Baumslag and Levin [2]; Eršov [5]; Hodges [9], [10]; Mal′cev [14]; Olin [16] and Saracino [19], [20].) In it we investigate the conditions under which the free product in the variety of all nilpotent of class 2 (nil-2) groups preserves saturation and stability.

It is well known that the direct product preserves both saturation (see Waszkiewicz and Wȩglorz [23]) and stability (see Wierzejewski [24]; Macintyre [13]; Eklof and Fisher [4]). On the other hand it is easy to show that the full free product of groups preserves neither property; indeed, in the case of saturation this failure is extremely bad since no free product of nontrivial groups is even 2-saturated. Our results show that the nil-2 free product falls between these two extremes.

Our proofs are mainly model-theoretic with a smattering of elementary algebra and rely heavily upon the unique normal form for the elements of a nil-2 free product given by MacHenry in [12]. (This normal form and some of its consequences are discussed in §1.) We assume familiarity with the basic ideas of saturation (see Chapter 5 of [3]) and Shelah's treatment of stability in [22].

We prove two main theorems in §3 each giving a necessary and sufficient condition in separate situations for the preservation of saturation. In the first (Theorem 3.1) we allow one finite factor, while in the second (Theorem 3.10) we deal solely with torsion groups. Our motivation for the proof of sufficiency was the paper of Waszkiewicz and Wȩglorz [23] and the principal tool is a “Feferman-Vaught” Theorem for the nil-2 free product which we prove in §2. We also show that if both factors in a nil-2 free product are nontorsion and one factor has a nil-2 basis, then the group is not even 3-saturated. We leave open the case where both factors are infinite but only one is torsion.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 3 , September 1981 , pp. 499 - 512
Copyright
Copyright © Association for Symbolic Logic 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

[1]Baldwin, J. T. and Saxl, J., Logical stability in group-theory, Journal of the Australian Mathematical Society (Series A), vol. 21 (1976), pp. 267–276.CrossRefGoogle Scholar
[2]Baumslag, B. and Levin, F., Algebraically closed torsion-free nilpotent groups of class 2, Communications in Algebra, vol. 4 (6)(1976), pp. 533–560.CrossRefGoogle Scholar
[3]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[4]Eklof, P. C. and Fisher, E. R., The elementary theory of abelian groups, Annals of Mathematical Logic, vol. 4 (1972), pp. 115–171.CrossRefGoogle Scholar
[5]Eršov, Ju. L., Theories of non-abelian varieties of groups, Proceedings of the Tarski Symposium, American Mathematical Society, Providence, R. I., 1974, pp. 255–264.CrossRefGoogle Scholar
[6]Feferman, S. and Vaught, R. L., The first order properties of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
[7]Fuchs, L., Abelian groups, Pergamon Press, Oxford, 1960.Google Scholar
[8]Hall, M., The theory of groups, Macmillan, New York, 1959.Google Scholar
[9]Hodges, W., A nullstellensatz for nilpotent groups (preprint).Google Scholar
[10]Hodges, W., Interpreting number theory in nilpotent groups (preprint).Google Scholar
[11]Hodges, W., private communication, 01 1978.Google Scholar
[12]MacHenry, T., The tensor product and the 2nd nilpotent product of groups, Mathematische Zeitschrift, vol. 73 (1960), pp. 134–145.CrossRefGoogle Scholar
[13]Macintyre, A., On ω1-categorical theories of abelian groups, Fundamenta Mathematicae, vol. 70 (1971), pp. 253–270.CrossRefGoogle Scholar
[14]Mal′cev, A. I., A correspondence betweenrings and groups, The metamathematics of algebraic systems, collected papers: 1936-1967, North-Holland, Amsterdam, 1971, pp. 124–137.Google Scholar
[15]Neumann, H., Varieties of groups, Springer-Verlag, New York, 1967.CrossRefGoogle Scholar
[16]Olin, P., Elementary properties of V-free products of groups, Journal of Algebra, vol. 47(1977), pp. 105–114.CrossRefGoogle Scholar
[17]Rogers, P. K., Topics in the model theory of abelian and nilpotent groups, Ph. D. Thesis, University of London, 04 1977.Google Scholar
[18]Sabbagh, M. G., Catégoricité et stabilité: quelques exemples parmi les groupes et anneaux, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Séries A et B, vol. 280 (1975), pp. 603–606.Google Scholar
[19]Saracino, D., Existentially complete nilpotent groups, Proceedings of the Abraham Robinson Memorial Conference, Yale University, May 1975, Israel Journal of Mathematics, vol. 25 (1976), pp. 241–248.Google Scholar
[20]Saracino, D., Existentially complete torsion-free nilpotent groups, this Journal, vol. 43 (1978), pp. 126–134.Google Scholar
[21]Scott, W. R., Group theory, Prentice-Hall, Englewood Cliffs, N. J., 1964.Google Scholar
[22]Shelah, S., Stability, the f.c.p. and superstability, Annals of Mathematical Logic, vol. 3 (1971), pp. 271–362.CrossRefGoogle Scholar
[23]Waszkiewicz, J. and Wȩglorz, B., On ω0-categoricity of powers, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 17 (1969), pp. 195–199.Google Scholar
[24]Wierzejewski, J., On stability and products, Fundamenta Mathematicae, vol. 93 (1976), pp. 81–95.CrossRefGoogle Scholar