Some Algorithms for the Calculation of Conjugacy Classes in the Sylow p-Subgroups of GL(n, q)

AbstractThe 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 = pt 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 n, n ≤ 8, we find the conjugacy vector and the number of conjugacy classes as polynomials in q of degree depending on n. Furthermore, for n ≤ 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