Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Cyclicity conditions for division algebras of prime degree

Authors: M. Mahdavi-Hezavehi and J.-P. Tignol
Journal: Proc. Amer. Math. Soc. 131 (2003), 3673-3676
MSC (2000): Primary 16K20
Published electronically: February 26, 2003
MathSciNet review: 1998187
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $D$ be a division algebra of prime degree $p$. A set of criteria is given for cyclicity of $D$in terms of subgroups of the multiplicative group $D^*$ of $D$. It is essentially shown that $D$ is cyclic if and only if $D^*$contains a nonabelian metabelian subgroup.

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

  • 1. S. A. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc., 80 (1955), 361-386. MR 17:577c
  • 2. P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series, vol. 81, Cambridge University Press, Cambridge, 1983. MR 696937
  • 3. I. N. Herstein, Finite multiplicative subgroups in division rings, Pacific J. Math. 3 (1953), 121-126. MR 14:1056j
  • 4. Hua Loo-Keng, Some properties of a field, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 533-537. MR 11:155c
  • 5. Hua Loo-Keng, On the multiplicative group of a field, Acad. Sinica Science Record 3 (1950), 1-6 (English, Chinese summary). MR 12:584e
  • 6. M. Š. Huzurbazar, The multiplicative group of a division ring, Soviet Math. Dokl. 1 (1960), 433–435. MR 0120252
  • 7. M. Mahdavi-Hezavehi, Free subgroups in maximal subgroups of 𝐺𝐿₁(𝐷), J. Algebra 241 (2001), no. 2, 720–730. MR 1843321,
  • 8. W. R. Scott, On the multiplicative group of a division ring, Proc. Amer. Math. Soc. 8 (1957), 303-305. MR 18:788g
  • 9. J.-P. Tignol, Sur les décompositions des algèbres à division en produit tensoriel d’algèbres cycliques, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin-New York, 1982, pp. 126–145 (French). MR 657427

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16K20

Retrieve articles in all journals with MSC (2000): 16K20

Additional Information

M. Mahdavi-Hezavehi
Affiliation: Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11365–9415, Tehran, Iran

J.-P. Tignol
Affiliation: Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

Received by editor(s): June 19, 2002
Received by editor(s) in revised form: July 15, 2002
Published electronically: February 26, 2003
Additional Notes: The first author thanks the Research Council of Sharif University of Technology for support. He also thanks Professor J.-P. Tignol for his hospitality during his stay at the Université Catholique de Louvain in May 2002.
The second author was partially supported by the National Fund for Scientific Research (Belgium).
Communicated by: Martin Lorenz
Article copyright: © Copyright 2003 American Mathematical Society