The exact number of conjugacy classes of the Sylow p-subgroups of GL(n, q) modulo (q-1)(13)

摘要

Let G be a finite p-group of order p(n). A well known result of P. Hall determines the number of conjugacy classes of G, r(G), modulo (p(2) - 1)(P - 1). Namely, he proved the existence of a non-negative constant k such that r(G) = n(p(2) - 1) + p(e) + k(p(2) - 1)(P - 1). We denote by g(n) the group of the upper unitriangular matrices over F-q, the finite field with q = p(t) elements. In [A. Vera-Lopez, J.M. Arregi, F.J. Vera-Lopez, On the number of conjugacy classes of the Sylow p-subgroups of GL(n, q). Bull. Austral. Math. Soc. 53 (1996) 431-439] the number (g(n),) is given modulo (q - 1)(5). In this paper, we introduce the concept of primitive canonical matrix. The knowledge of the number of primitive canonical matrices with connected graph of size less than or equal to n should be sufficient to determine the number of all canonical matrices of size n. Moreover, we give explicitly the polynomial formulas mu(i) = mu(i) (n), i = 0,..., 12, depending only on n, and not on q, such that r(g(n)) = Sigma(i)mu(i)(n)(q - 1)(i) + k (n, q) (q - 1)(13) for all n is an element of N.