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Let G be the group of all 2 x 2 matrices

[tex]

\begin{pmatrix}

a & b \\ c & d

\end{pmatrix}

[/tex]

where a, b, c, d are integers modulo p, p a prime number, such that ad - bc ≠ 0. G forms a group relative to matrix multiplication. What is the order of G?

The attempt at a solution

Let's consider the case p = 3. According to the book, G has 48 elements. I don't see how this is possible. The possible values for a, b, c, d are 0, 1, 2, so there are 3 * 3 * 3 * 3 = 81 possible matrices. The ones we don't want are those that satisfy ad = bc and there are 3 + (3 * 2) * 2 = 15 of these. There should be 81 - 15 = 66 matrices in G. Right?

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# Homework Help: Order of Group of 2 x 2 Matrices

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