Recall the definitions of the multiplicative groups

Question

Problems:

a) Recall the following definitions of the multiplicative groups GLn(k) and SLn(k) over a field k:

                        GLn(k)={invertible n x n matrices over k}
                        SLn(k)={A in GLn(k) such that the determinant of A=1}

Prove that SLn(k) is a normal subgroup of GLn(k) and that the quotient group GLn(k)/SLn(k) is isomorphic to the multiplicative group k*={a in k such that a is not equal to zero}.

b) Determine the number of elements in the finite group GL3(Zp)