Show that the subgroup of strictly upper triangular matrices (i.e. with only 1 in diagonal entries) in is a Sylow -subgroup.
We know that the order of is . Note that does not divide for any . Moreover, we have . Thus (by definition) a Sylow -subgroup of has elements.
Recall that a matrix is strictly upper triangular if all diagonal entries are 1, all lower entries are 0, and all upper entries can be any field element. Then the number of elements in is , where is the number of entries above the main diagonal. We can see that this number is . Thus is a Sylow -subgroup in .