Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T06:21:46.146Z Has data issue: false hasContentIssue false
  • English
  • Français

Logical stability in group theory

Published online by Cambridge University Press:  09 April 2009

J. T. Baldwin  and
Jan Saxl
J. T. Baldwin
Affiliation:
Department of Mathematics, University of Illionis, at Chicago Circle Box 4348 Chicago, Illinois 60680, U.S.A.
Jan Saxl
Affiliation:
Department of Mathematics, University of Illionis, at Chicago Circle Box 4348 Chicago, Illinois 60680, U.S.A.
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 paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety of groups is unstable.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 3 , May 1976 , pp. 267 - 276
Copyright
Copyright © Australian Mathematical Society 1976

References

Baldwin, J. T. and Lachlan, A. H. (1973), ‘On universal Horn classes categorical in some infinite power’, Algebra Universalis, 3, 98–111.CrossRefGoogle Scholar
Berthier, D. (1973), ‘Stability, Products, Groups’, Notices Amer. Math. Soc. 73T-E98.Google Scholar
Eklof, P. and Fisher, E. (1972), ‘The elementary theory of Abelian groups’, Annals Math. Log. 4, 115–171.CrossRefGoogle Scholar
Keisler, H. J. (1974), ‘The number of types in a first order theory’, Notices Amer. Math. Soc. 74T-E8 p. A316. Keisler 1975.Google Scholar
Keisler, H. J. (1975), ‘Six Classes of Theories’, J. Austral. Math. Soc. 21 (Series A), 129–138.Google Scholar
Lachlan, A. H. (1974), ‘Two conjectures regarding ω-stable theories’, Fund. Math. 81, 133–145.CrossRefGoogle Scholar
Macintyre, A. (1970), ‘On ω1-categorical theories of Abelian groups’, Fund. Math. 70, 253–270.CrossRefGoogle Scholar
McKenzie, R. (1970), ‘On elementary types of symmetric groups’, Alg. Univ. 1, 13–21.CrossRefGoogle Scholar
Morley, M. (1965), ‘Categoricity in power’, Trans. Amer. Math. Soc. 114, 514–538.CrossRefGoogle Scholar
Rosenstein, J. G. (1973), ‘N0-categoricity of Groups’, J. of Algebra 25, 435–467.CrossRefGoogle Scholar
Sabbagh, G. (1975a), ‘Logique Mathematique.—Categoricité et Stabilité: construction les préservant et conditions de chaine’, C. R. Acad. Sc. Paris, t. 280 (3 03 1975).Google Scholar
Sabbagh, G. (1975b), ‘Logique Mathematique.—Categoricité et stabilité: quelques examples parmi les groupes et anneaux’, C. R. Acad. Sc. Paris, t. 280 (10 03 1975).Google Scholar
Sacks, G. E. (1972), ‘Saturated Model Theory’, W. A. Benjamin Inc., Reading, Mass.Google Scholar
Scott, W. R. (1951), ‘Algebraically Closed Groups’, Proc. Amer. Math. Soc. 2, 118–121.CrossRefGoogle Scholar
Shelah, S. (1969), ‘Stable theories’, Israel J. Math. 7, 187–202.CrossRefGoogle Scholar
Shelah, S. (1971a), ‘Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory’, Annals Math. Log. 3, 271–362.CrossRefGoogle Scholar
Shelah, S. (1971b), ‘The number of non-isomorphic models of an unstable first-order theory’, Israel J. Math., 9, 473–487.CrossRefGoogle Scholar
Szmielew, W. (1955), ‘Elementary properties of Abelian groups’, Fund. Math. 41, 203–271.CrossRefGoogle Scholar