ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 Nicolas, Jean-Louis, Université Claude Bernard, Lyon, France On the maximum order of torsion elements in GL(n, ℤ) The maximum order g(n) of an element of the symmetric group S_{n} was first studied by E. Landau in 1903 who proved

log g(n) ∼ \sqrt{n log n}

as n → +∞. A torsion element in GL(n, ℤ) is a matrix M with integral coefficients, n rows and n colomns, and a finite order e such that M^{e} = I, the unit matrix. Let us denote by G(n) the maximum order of these torsion elements. It is easy to see that g(n) ≤ G(n). In [1], with G. Levitt, we have proved

log G(n) ∼ \sqrt{n log n}

and some other properties of G(n), for instance, the largest prime number dividing G(n) is asymptotic to \sqrt{n log n}, as n → +∞. Here, I shall prove the following result: Theorem. As n → +∞, one has G(n)/g(n) → +∞, and more precisely,

(log n)^{1+o(1)} ≤ \frac{G(n)}{g(n)} ≤ exp((log log n)^{2+o(1)}).

Under Riemann hypothesis, the upper bound can be replaced by (log n)^{3+o(1)}. References: [1] G. Levitt and J. -L. Nicolas, On the maximum order of torsion elements in GL(n, ℤ) and Aut(F_{n}), to appear, J. of Algebra.