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Multipliciative groups

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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)

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This shows how to complete a proof regarding a multiplicative group and determine the number of elements ina given finite group.

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(1) We view matrix groups with respect to their multiplicative structures. Consider the homomorphism det : GLn(k) → k ? . Notice that ker(det) = {X ∈ GLn(k) : det X = 1} = SLn(k) As the kernel of a group homomorphism, SLn(k) is a normal subgroup of GLn(k). Notice also that the image of det consists of all the nonzero elements of k, since an invertible matrix has nonzero determinant, and every nonzero scalar (i.e., an element of k ? ) can be realized as ...

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