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Representations of infinite permutations
by words (II)

Authors: Randall Dougherty and Jan Mycielski
Journal: Proc. Amer. Math. Soc. 127 (1999), 2233-2243
MSC (1991): Primary 20F05
Published electronically: April 9, 1999
MathSciNet review: 1605952
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Abstract | References | Similar Articles | Additional Information

Abstract: We present an argument (due originally to R. C. Lyndon) which completes the proof of the following theorem: Every free group word which is not a proper power can represent any permutation of an infinite set.

References [Enhancements On Off] (What's this?)

  • 1. J. L. Brenner, R. J. Evans, and D. M. Silberger, The universality of words $x^{r} y^{s}$ in alternating groups, Proc. Amer. Math. Soc. 96 (1986), 23-28. MR 88a:20047
  • 2. R. Dougherty, Products of two cycles (91-6(b)) - solution, in Mathematical Entertainments column, D. Gale, ed., Math. Intelligencer 16 (2) (1994), 30-31.
  • 3. M. Droste, Classes of universal words for the infinite symmetric group, Algebra Universalis 20 (1985), 205-216. MR 87d:20049
  • 4. -, On the universality of words for the alternating groups, Proc. Amer. Math. Soc. 96 (1986), 18-22. MR 87c:20063
  • 5. A. Ehrenfeucht, S. Fajtlowicz, J. Malitz, and J. Mycielski, Some problems on the universality of words in groups, Algebra Universalis 11 (1980), 261-263. MR 81k:20045
  • 6. D. Gale, Mathematical entertainments, Math. Intelligencer 15 (3) (1993), 56-61.
  • 7. R. C. Lyndon, Words and infinite permutations, Mots, Lang. Raison Calc., Hermès, Paris, 1990, pp. 143-152. MR 95c:20006
  • 8. J. Mycielski, Can one solve equations in groups?, Amer. Math. Monthly 84 (1977), 723-726.
  • 9. -, Equations unsolvable in $GL_{2}(C)$ and related problems, Amer. Math. Monthly 85 (1978), 263-265. MR 57:9867
  • 10. -, Representations of infinite permutations by words, Proc. Amer. Math. Soc. 100 (1987), 237-241. MR 88c:20044
  • 11. O. Ore, Some remarks on commutators, Proc. Amer. Math. Soc. 2 (1951), 307-314. MR 12:671e
  • 12. D. M. Silberger, Are primitive words universal for infinite symmetric groups?, Trans. Amer. Math. Soc. 276 (1983), 841-852. MR 84c:20011
  • 13. C. M. Weinbaum, On relators and diagrams for groups with one defining relation, Illinois J. Math. 16 (1972), 308-322. MR 45:6901

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Additional Information

Randall Dougherty
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Jan Mycielski
Affiliation: Department of Mathematics, University of Colorado, Boulder, Colorado 80309

Received by editor(s): November 1, 1997
Published electronically: April 9, 1999
Additional Notes: The first author was supported by NSF grant number DMS-9158092 and by a fellowship from the Sloan Foundation.
Dedicated: Dedicated to the memory of Roger C. Lyndon
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1999 American Mathematical Society

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