Conjugacy classes in Sylow p-subgroups of GL(n, q)

Involutions and characters of upper triangular matrix groups

“…A result highlighting this difference can be obtained by combining work of J. Arregi and A. Vera-López [8,9] with work of the present authors [5,6].…”

“…The computation in [9] was made possible by the fact that it suffices to check just one element in each conjugacy class. The 2 66 elements of U (12) are divided into only 34, 064, 872 classes, and according to [8], each class contains a matrix in an appropriate canonical form.…”

Conjugacy classes in sylow p-subgroups of GL(n, q), IV

“…We note, using Lemma 3.9 of [3], that the entries of the matrix A r<s which are not placed over pivots are ramification points. This also proves that the matrix is canonical, since all the non-zero values correspond to ramification points.…”

“…Here, we obtain the third component, that is, the number of conjugacy classes whose centralizer has q" +l elements. Besides, we give the whole set of numbers which compose this vector:We keep the definitions and notations of [2][3][4]. We recall that an element a tJ of a matrix A e ©" is a pivot if it is the first nonzero element in its row, out of the main diagonal, that is a iJc = 0 for k = i + 1 , .…”

“…We keep the definitions and notations of [2][3][4]. We recall that an element a tJ of a matrix A e ©" is a pivot if it is the first nonzero element in its row, out of the main diagonal, that is a iJc = 0 for k = i + 1 , .…”