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Algebra and Logic

, Volume 58, Issue 1, pp 59–76 | Cite as

Generating Triples of Involutions of Groups of Lie Type of Rank 2 Over Finite Fields

  • Ya. N. NuzhinEmail author
Article
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For finite simple groups U5(2n), n > 1, U4(q), and S4(q), where q is a power of a prime p > 2, q − 1 ≠= 0(mod4), and q ≠= 3, we explicitly specify generating triples of involutions two of which commute. As a corollary, it is inferred that for the given simple groups, the minimum number of generating conjugate involutions, whose product equals 1, is equal to 5.

Keywords

group of Lie type finite simple group generating triples of involutions 

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References

  1. 1.
    Ya. N. Nuzhin, “Generating triples of involutions for alternating groups,” Mat. Zametki, 51, No. 4, 91-95 (1992).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ya. N. Nuzhin, “Generating triples of involutions for Chevalley groups over a finite field of characteristic 2,” Algebra and Logic, 29, No. 2, 134-143 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ya. N. Nuzhin, “Generating triples of involutions for Lie-type groups over a finite field of odd characteristic. I,” Algebra and Logic, 36, No. 1, 46-59 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ya. N. Nuzhin, “Generating triples of involutions for Lie-type groups over a finite field of odd characteristic. II,” Algebra and Logic, 36, No. 4, 245-256 (1997).Google Scholar
  5. 5.
    V. D. Mazurov, “On generation of sporadic simple groups by three involutions two of which commute,” Sib. Math. J., 44, No. 1, 160-164 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford (1985).Google Scholar
  7. 7.
    Unsolved Problems in Group Theory, The Kourovka Notebook, No. 19, Institute of Mathematics SO RAN, Novosibirsk (2018); http://www.math.nsc.ru/alglog/19tkt.pdf.
  8. 8.
    J. M. Ward, Generation of simple groups by conjugate involutions, PhD Thesis, Queen Mary college, Univ. London (2009).Google Scholar
  9. 9.
    E. S. Rapaport, “Cayley color groups and Hamilton lines,” Scripta Math., 24, 51-58 (1959).MathSciNetzbMATHGoogle Scholar
  10. 10.
    I. Pak and R. Radoiˇci´c, “Hamiltonian paths in Cayley graphs,” Discr. Math., 309, No. 17, 5501-5508 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    G. A. Jones, “Automorphism groups of edge-transitive maps,” arXiv:1605.09461 [math.CO].Google Scholar
  12. 12.
    M. Maˇcaj, “On minimal kaleidoscopic regular maps with trinity symmetry,” The Seventh Workshop Graph Embeddings and Maps on Surfaces, Abstracts, Podbanske, Slovakia (2017).Google Scholar
  13. 13.
    R. W. Carter, Simple Groups of Lie Type, Pure Appl. Math., 28, Wiley, London (1972).Google Scholar
  14. 14.
    L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory, B. G. Teubner, Leipzig (1901).zbMATHGoogle Scholar
  15. 15.
    D. Gorenstein, Finite Groups, Harper and Row, New York (1968).Google Scholar
  16. 16.
    Ya. N. Nuzhin, “Generating sets of elements of Chevalley groups over a finite field,” Algebra and Logic, 28, No. 6, 438-449 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    V. M. Levchuk, “Remark on a theorem of L. Dickson,” Algebra and Logic, 22, No. 4, 306-316 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ya. N. Nuzhin, “Groups contained between groups of Lie type over various fields,” Algebra and Logic, 22, No. 5, 378-389 (1983).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Siberian Federal UniversityKrasnoyarskRussia

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