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Let [tex] p [/tex] be a prime. Prove that the order of [tex] GL_2 ( \mathbb{Z} / p \mathbb{Z} ) [/tex] is [tex] p^{4} - p^{3} - p^{2} + p [/tex]

The text suggests subtracting the number of 2 x 2 matrices which are not invertible from the total number of 2 x 2 matrices over [tex] \mathbb{Z} / p \mathbb{Z} [/tex]

I have been working on this for awhile, but it's not going well.

First, it seems obvious to me that the total number of 2 x 2 matrices over [tex] \mathbb{Z} / p \mathbb{Z} [/tex] must be [tex] p^{4} [/tex] since each of the 4 entries has [tex] p [/tex] possible values.

Based on this assumption, I am forced to conclude that there are [tex] p^{3} + p^{2} - p [/tex] of these matrices that are not invertible. However, I'm having a hard time showing that this is true, if indeed it is.

Any help is greatly appreciated.

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# Homework Help: General Linear Group Order

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