SOME ALGORITHMS FOR THE CALCULATION OF CONJUGACY CLASSES IN THE SYLOW P-SUBGROUPS OF GL(N,Q)

The aim of this paper is to fix theoretical bases and develop algorithms for the symbolic calculation of the conjugacy classes in the Sylow p-subgroup of GL(n, q) formed by the upper unitriangular matrices, q = p(t) being arbitrary. Our main method is to go back along a central series whose factor groups correspond to the entries in the upper diagonals, according to a suitable order. By linearizing the problem, we are able to construct representatives of the conjugacy classes. For small values of it, n less than or equal to 8, we find the conjugacy vector and the number of conjugacy classes as polynomials in q of degree depending on it. Furthermore, for n less than or equal to 7 we observe that the distinction between zero and non-zero values in the entries suffices to construct a complete set of representatives of the conjugacy classes. (C) 1995 Academic Press, Inc.