Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-19T19:05:38.288Z Has data issue: false hasContentIssue false
  • English
  • Français

How to Construct Almost Free Groups

Published online by Cambridge University Press:  20 November 2018

Alan H. Mekler
Alan H. Mekler*
Affiliation:
University of Toronto, Toronto, Ontario
Rights & Permissions [Opens in a new window]

Extract

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.

Almost free groups were introduced in [9] as groups all of whose “small” subgroups are free. Here “small” means generated by fewer elements than the cardinality of the group. This concept is a generalization of locally free. Suppose κ is a cardinal > ω. A group is κ-free if every subgroup generated by fewer than κ elements is free. A group of cardinality κ which is κ-free is almost free. There are two related concepts which are closer approximations to freeness.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1206 - 1228
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Baumgartner, J. E., Notes from the Cambridge Summer School in Set Theory (1978).Google Scholar
2. Baumslag, G., Groups with the same lower central sequence as a relatively free group. I The groups, Trans. Amer. Math. Soc. 129 (1967), 308–321.Google Scholar
3. Eklof, P. C., Infinitary equivalence of abelian groups, Fund. Math. 81 (1974), 305–314.Google Scholar
4. Eklof, P. C., On the existence of κ-free abelian groups, Proc. Amer. Math. Soc. 47 (1975), 65–72.Google Scholar
5. Eklof, P. C., Whitehead's problem is undecidable, Amer. Math. Monthly 83 (1976), 775–778.Google Scholar
6. Eklof, P. C., Methods of logic in abelian group theory, in Abelian Group Theory, Springer Verlag Lecture Notes in Math, no 616 (1977), 251–269.CrossRefGoogle Scholar
7. Eklof, P. C. and Mekler, A. H., On constructing indecomposable groups in L, J. Algebra 40 (1977), 96–103.Google Scholar
8. Eklof, P. C. and Mekler, A. H., Infinitary stationary logic and abelian groups, (to appear).CrossRefGoogle Scholar
9. Higman, G., Almost free groups, Proc. London Math. Soc. 1 (1951), 284–290.Google Scholar
10. Hill, P., New criteria for freeness in abelian groups II, Trans. Amer. Math. Soc. 196 (1974), 191–201.Google Scholar
11. Jensen, R., The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308.Google Scholar
12. Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory (Interscience, New York, 1966).Google Scholar
13. Mekler, A. H., Applications of logic to group theory, Ph.D. thesis, Stanford University (1976).Google Scholar
13. Mekler, A. H., The number of n-free groups and the size of Ext., in Abelian Group Theory, Springer Verlag Lecture Notes in Math, no 616 (1977), 323–331.Google Scholar
15. Mitchell, W., Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5 (1972), 21–46.Google Scholar
16. Nadel, M. and Stavi, J., L∞λ-equivalence, isomorphism and potential isomorphism, Trans. Amer. Math. Soc. 236 (1978), 51–74.Google Scholar
16. Shelah, S., Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 18 (1974), 243–256.Google Scholar
18. Shelah, S., A compactness theorem for singular cardinals, free algebra, Whitehead problem and transversals, Israel J. Math. 21 (1975), 319–349.Google Scholar
19. Solvay, R., Real-valued measurable cardinals, in Axiomatic Set Theory, A.M.S. Proc. of Symp. in Pure Math XIII (Part 1), Providence (1971), 397–428.Google Scholar