Israel Journal of Mathematics

, Volume 91, Issue 1–3, pp 349–371 | Cite as

The order of a typical matrix with entries in a finite field

  • Eric Schmutz
Article

Abstract

IfA is an invertiblen×n matrix with entries in the finite field Fq, letT n (A) be its minimum period or exponent, i.e. its order as an element of the general linear group GL(n,q). The main result is, roughly, that\(T_n (A) = q^{n - } (log n)^{2 + 0(1)} \) for almost everyA.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Arratia, A. D. Barbour and S. Tavaré,On random polynomials over finite fields, Mathematical Proceedings of the Cambridge Philosophical Society114 (1993), 347–368.MATHMathSciNetGoogle Scholar
  2. [2]
    R. Arratia and S. Tavaré,Limit theorems for combinatorial structures via discrete process approximations, Random Structures and Algorithms3 (1992), 321–345.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    R. Arratia and S. Tavaré,Independent process approximations for random combinatorial structures, Advances in Mathematics104 (1994), 90–154.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    M. Car,Factorisation dans F q[x], Comptes Rendus de l'Academie des Sciences, Paris294 (1982), 147–150.MATHMathSciNetGoogle Scholar
  5. [5]
    J. Curtiss,A note on the theory of moment generating functions, Annals of Mathematical Statistics13 (1942), 430–433.MathSciNetMATHGoogle Scholar
  6. [6]
    P. Erdös, C. Pomerance and E. Schmutz,Carmichael's lambda function, Acta ArithmeticaLVIII.4 (1991), 363–385.Google Scholar
  7. [7]
    P. Erdős and P. Turán,On some problems of a statistical group theory III, Acta Math. Acad. Sci. Hung.18 (1967), 309–320.CrossRefGoogle Scholar
  8. [8]
    P. Erdős On the sum\(\sum {_{d|2^n - 1} } \frac{1}{d}\), Israel Journal of Mathematics9 (1971), 43–48.MathSciNetGoogle Scholar
  9. [9]
    P. Flajolet and A. M. Odlyzko,Singularity analysis of generating functions, SIAM Journal of Discrete Mathematics3 (1990), 216–240.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    P. Flajolet and M. Soria,General combinatorial schemas with Gaussian limit distributions and exponential tails, Discrete Mathematics114 (1993), 159–180.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    P. Flajolet and M. Soria,Gaussian limiting distributions for the number of components in combinatorial structures, Journal of Combinatorial Theory, Series A53 (1990), 165–182.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Z. Gao and L. B. Richmond,Central and local limit theorems applied to asymptotic enumertion IV: multivariate generating functions, Journal of Computational and Applied Mathematics41 (1992), 177–186.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Gerstenhaber,On the number of nilpotent matrices with coefficients in a finite field, Illinois Journal of Mathematics5 (1961), 330–336.MATHMathSciNetGoogle Scholar
  14. [14]
    W. Goh and E. Schmutz,A Central Limit, Theorem on GL n (Fq), Random Structures and Algorithms2 (1991), 47–53.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    R. R. Hall and G. Tenenbaum,Divisors, Cambridge University Press, 1988.Google Scholar
  16. [16]
    J. C. Hansen,Factorization in GF(q m)[x] and Brownian motion, Combinatorics, Probability, and Computing2 (1993), 285–299.MATHMathSciNetGoogle Scholar
  17. [17]
    J. C. Hansen and E. Schmutz,How random is the characteristic polynomial of a random matrix? Mathematical Proceedings of the Cambridge Philosophical Society114 (1993), 507–516.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    M. Kac,Statistical independence in probability, analysis and number theory, Carus Monographs No. 12, MAA, 1959, p. 56.Google Scholar
  19. [19]
    P. Kiss and B. M. Phong,On a problem of Rotkiewicz, Mathematics of Computation48 (1987), 751–755.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    V. Kolchin,Random Mappings, Optimization Software Inc., New York, 1986.MATHGoogle Scholar
  21. [21]
    R. Lidl and H. Niederreiter,Finite Fields, Encyclopedia of Mathematics and Its Applications20, Addison Wesley, 1983.Google Scholar
  22. [22]
    A. M. Odlyzko,Discrete logarithms in finite fields and their cryptographic signifigance, Proceedings of Eurocrypt '84, Springer Lecture Notes in Computer Science209 (1984), 225–314.Google Scholar
  23. [23]
    R. Stong,The average order of a matrix, Journal of Combinatorial Theory, Series A64 (1993), 337–343.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    R. Stong,Some asymptotic results on finite vector spaces, Advances in Applied Mathematics9 (1988), 167–199.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    H. Wilf,Generatingfunctionology, Academic Press New York, 1990.MATHGoogle Scholar

Copyright information

© The Magnes Press 1995

Authors and Affiliations

  • Eric Schmutz
    • 1
  1. 1.Mathematics and Computer Science DepartmentDrexel UniversityPhiladelphiaUSA

Personalised recommendations