Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-20T04:13:19.528Z Has data issue: false hasContentIssue false
  • English
  • Français

A Short Combinatorial Proof of the Vaught Conjecture

Published online by Cambridge University Press:  20 November 2018

Charles C. Edmunds
Charles C. Edmunds*
Affiliation:
University of ManitobaWinnipegManitoba
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.

In [5] R. C. Lyndon gave the first proof of the Vaught conjecture: that if a, b9 and c are elements of a free group F such that a2b2=c2, then ab=ba. Lyndon's proof has been followed by many alternative proofs and generalizations [1, 2, 3, 4, 6, 8, 9, 10, 11, 13, 14] all of which involve rather long combinatorial arguments or group theoretical arguments of a noncombinatorial nature.This note provides a short, purely combinatorial proof of the Vaught conjecture.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 4 , October 1975 , pp. 607 - 608
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Baumslag, G., On a problem of Lyndon, J. London Math. Soc. 35 (1960), 30–32.Google Scholar
2. Baumslag, G., Residual nilpotence and relations in free groups, J. Algebra 2 (1965), 271–282.Google Scholar
3. Baumslag, G. and Steinberg, A., Residual nilpotence and relations in free groups, Bull. Amer. Math. Soc. 70 (1964), 283–284.Google Scholar
4. Budnika, L. G. and Markov, Al. A., On F-semigroups with three generators, Math. Notes, 14(1974), 711–716.Google Scholar
5. Lyndon, R. C., The equation a2b2=c2 in free groups, Michigan Math. J. 6 (1959), 89–95.Google Scholar
6. Lyndon, R. C. and Schützenberger, M. P., The equation aM=bNcP in a free group, Michigan Math. J. 9 (1962), 289–298.Google Scholar
7. Magnus, W., Karrass, A., and Solitar, D., Combinatorial Group Theory, Pure and Applied Math. Vol. 13, Interscience, New York, 1966.Google Scholar
8. Schutzenberger, M. P., Sur l’équation a2+n=b2+mc2+p dans un group libre, C.R. Acad. Sci. Paris 248 (1959), 2435–2436.Google Scholar
9. Shenkman, E., The equation anhn=cn in a free group, Ann. of Math. (2) 70 (1959), 562–564.Google Scholar
10. Stallings, J., On certain relations in free groups, Notices Amer. Math. Soc. 6 (1959), 532.Google Scholar
11. Steinberg, A., Ph.D. thesis, N.Y.U., 1962.Google Scholar
12. Wicks, M. J., Commutators in free products, J. London Math. Soc. 37 (1962), 433–444.Google Scholar
13. Wicks, M. J., A general solution of binary homogenous equations over free groups, Pacific J. Math. 41 (1972), 543–561.Google Scholar
14. Wicks, M. J., A relation in free products, Proc. Second Internat. Conf. Theory of Groups, Canberra, 1973, 709–716.Google Scholar