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Finite groups embeddable in division rings


Author: T. Y. Lam
Journal: Proc. Amer. Math. Soc. 129 (2001), 3161-3166
MSC (2000): Primary 12E15, 16Kxx, 20B05; Secondary 20D20, 20B07, 16U60
DOI: https://doi.org/10.1090/S0002-9939-01-05961-5
Published electronically: April 17, 2001
MathSciNet review: 1844988
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Abstract | References | Similar Articles | Additional Information

Abstract: In a tour de force in 1955, S. A. Amitsur classified all finite groups that are embeddable in division rings. In particular, he disproved a conjecture of Herstein which stated that odd-order emdeddable groups were cyclic. The smallest counterexample turned out to be a group of order 63. In this note, we prove a non-embedding result for a class of metacyclic groups, and present an alternative approach to a part of Amitsur’s results, with an eye to “de-mystifying" the order 63 counterexample.


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

  • S. A. Amitsur: Finite subgroups of division rings, Trans. Amer. Math. Soc. 80(1955), 361-386.
  • W. Burnside: Theory of Groups of Finite Order, 2nd edition, Cambridge U. Press, 1911. (Reprinted by Dover Publications, 1955.)
  • John Dauns, A concrete approach to division rings, R & E, vol. 2, Heldermann Verlag, Berlin, 1982. MR 671253
  • B. Fein and M. Schacher: Embedding finite groups in rational division algebras, I, II, J. Algebra 17(1971), 412-428, and 19(1971), 131-139. ;
  • Charles Ford, Finite groups and division algebras, Enseign. Math. (2) 19 (1973), 313–327. MR 342561
  • Marshall Hall Jr., The theory of groups, The Macmillan Co., New York, N.Y., 1959. MR 0103215
  • I. N. Herstein: Finite multiplicative subgroups of division rings, Pacific J. Math. 3(1953), 121-126.
  • Nathan Jacobson, Basic algebra. I, 2nd ed., W. H. Freeman and Company, New York, 1985. MR 780184
  • A. Shokrollahi: Packing unitary matrices, Colloquium talk, University of California, Berkeley, Calif., November, 2000.

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Additional Information

T. Y. Lam
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
MR Author ID: 109495
Email: lam@math.berkeley.edu

Received by editor(s): March 13, 2000
Published electronically: April 17, 2001
Communicated by: Lance W. Small
Article copyright: © Copyright 2001 copyright retained by the author