Number of elements in $GL_2(\mathbb Z_p)$

Asked Aug 14 '17 at 14:52

Active Aug 14 '17 at 14:52

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I want to find the number of elements in $GL_2(\mathbb Z_p)$, where $p$ is a prime.

The first row can be chosen in $p^2-1$ ways as we can not take both zeros. The second row must be chosen in such a way that it is not a linear combination of the first. Total number of rows those are linear combinations of the first row is $p$. Thus total possibilities for the second row is $p^2-p$. Thus $|GL_2(\mathbb Z_p)|=(p^2-1)(p^2-p)$. Am I right?

asked Aug 14 '17 at 14:52

Anupam

5,072

1

Is $\mathbb{Z}_p$ the field with $p$ elements or the $p$-adic integers? If it's the first (what seems to be the case) this is correct. – Verdruss Aug 14 '17 at 14:53
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Yes, you are right. – Randall Aug 14 '17 at 14:55
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That is correct. A similar argument shows that $|GL_n(\mathbb{Z}_p)| = (p^n-1)(p^n-p)\cdots(p^n-p^{n-1}).$ – Ken Duna Aug 14 '17 at 15:04

Number of elements in $GL_2(\mathbb Z_p)$

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