On cyclotomic schemes over finite near-fields
J. Bagherian
, Ilia Ponomarenko
and A. Rahnamai Barghi3
3I. Ponomarenko Petersburg Department of V.A. Steklov Institute of Mathematics, Fontanka 27, St. Petersburg 191023, Russia
DOI: 10.1007/s10801-007-0081-4
Abstract
We introduce a concept of cyclotomic association scheme over a finite near-field \mathbb K \mathbb{K} . It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL( V), where V is the linear space associated with \mathbb K \mathbb{K} . A sufficient condition on a cyclotomic scheme C \mathcal{C} that guarantee the inclusion Aut( C) \sterling A G L(1,\mathbb F), \mathrm{Aut}(\mathcal{C})\le \mathrm{A} Γ\mathrm{L}(1,\mathbb{F}), where \mathbb F \mathbb{F} is a finite field with |\mathbb K | |\mathbb{K}| elements, is given.
Pages: 173–185
Keywords: keywords association scheme; finite near-field; permutation group
Full Text: PDF
References
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2. Cossidente, A., & de Resmini, M. J. (2004). Remarks on Singer cyclic groups and their normalizers. Designs, Codes and Cryptography, 32, 97-102.
3. Dixon, J. D., & Mortimer, B. (1996). Permutation groups. Graduate texts in mathematics (Vol. 163). New York: Springer.
4. Guralnick, R., Penttila, T., Praeger, C. E., & Saxl, J. (1999). Linear groups with orders having certain large prime divisors. Proceedings of the London Mathematical Society, 78, 167-214.
5. Kleidman, P., & Liebeck, M. (1990). The subgroup structure of the finite classical groups. London mathematical society lecture note series (Vol. 129). Cambridge: Cambridge University Press.
6. Manz, O., & Wolf, T. R. (1993). Representations of solvable groups. London mathematical society lecture note series (Vol. 185). Cambridge: Cambridge University Press.
7. Neumann, P. M., & Praeger, C. E. (1992). A recognition algorithm for special linear groups. Proceedings of the London Mathematical Society, 65, 555-603.
8. Passman, D. S. (1968). p-Solvable doubly transitive permutation groups. Pacific Journal of Mathematics, 26, 555-577.
9. Praeger, C. E., & Saxl, J. (1992). Closures of finite primitive permutation groups. The Bulletin of the London Mathematical Society, 24, 251-258.
10. Ribenboim, P. (2000). My numbers, my friends. Popular lectures on number theory. New York: Springer.
11. Roitman, M. (1997). On Zsigmondy primes. Proceedings of the American Mathematical Society, 125, 1913-1919.
12. Taylor, D. E. (1992). The geometry of the classical groups. Berlin: Heldermann.
13. Wähling, H. (1987). Theorie der Fastkörper. Essen: Thales.
14. Wielandt, H. (1964). Finite permutation groups. New York: Academic.