Let g ut nf be the set of the upper triangular n x n

Question

Let G = UT (n,F) be the set of the upper triangular n x n matrices with entries in a field F with p elements and 1's on the diagonal. The operation in G is matrix multiplication.

(a) Show that G is a group

(b) Show that G is a finite p-group

(c) Consider the upper central series of G:
1 = Z_0 (G) <= Z_1 (G) <= Z_2 (G) <=....<= Z_c (G) = G

Write explicitly what Z_1 (G), Z_2 (G), and Z_3 (G) are.

Show that Z_3 / Z_2 = Z (G/Z_2)

Prove that G is nilpotent of class exactly n-1.