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

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 = 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.