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This question is from the CSIR NET, December 2023 test. Note that multiple options could be correct.

Let $G$ be the group (under matrix multiplication) of $2\times 2$ invertible matrices with entries from the ring $\mathbb Z/ 9 \mathbb Z$ . Let $a$ be the order of $G$. Which of the following statements is/are true?

  1. $a$ is divisible $3^4$.
  2. $a$ is divisible by $2^4$.
  3. $a$ is not divisible by $48$.
  4. $a$ is divisible by $3^6$.

I know how to compute the order of $\text{GL}(n,\mathbb F)$ when $\mathbb F$ is a finite field. There's an easy-to-understand formula which comes from linear independence of the columns and some elementary combinatorics.
However, I am not sure if there's a formula for the order of $\text{GL}(n,R)$ when $R$ is an arbitrary finite ring. The question does not ask for exact order of $\text{GL}(2,\mathbb Z/9\mathbb Z)$ so here's how I tried to attempt it:

Note that a matrix from $M_2(\mathbb Z/9\mathbb Z)$ is invertible if and only if its determinant is from the group of units $(\mathbb Z/9\mathbb Z)^*$ which has $6$ elements. Consider the subgroup of upper triangular matrices in $\text{GL}(2,\mathbb Z/9\mathbb Z)$. The diagonal entries need to be unit, the top left entry could be anything. Thus, the order of this subgroup is $6\times 6\times 9$. Using Lagrange's theorem, I can confirm that option $1$ is true since $3^4\mid 6\times 6\times 9\,\mid \, \text{#}\text{GL}(2,\mathbb Z/9\mathbb Z)$.

However, I can't decide for the other options. I need help.

Nothing special
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    I think the linear algebra tag would be appropriate – J. W. Tanner Oct 08 '24 at 19:18
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    For the formula, see here, in the comments. So $|GL_2(\Bbb Z/p^2\Bbb Z)|=p^4(p^2-1)(p^2-p)$. Your case is $p=3$. So 1. and 2. are true, but 3. and 4. are false. – Dietrich Burde Oct 08 '24 at 19:33
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    @DietrichBurde Thanks. I think I understand the comment by the user reuns. – Nothing special Oct 08 '24 at 19:53
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    Even better is the answer by Qiaochu Yuan in the duplicate for $GL_2(R)$ in general, where $R$ is a finite local ring. – Dietrich Burde Oct 08 '24 at 19:56
  • @DietrichBurde Thank you I didn't see that... I do not know why a useless answer is leading that page instead of the accepted one. – Nothing special Oct 08 '24 at 19:59
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    @Nothingspecial Underneath the question, you should see a drop down menu that says "Sorted by:". Make sure "Highest score" rather than "date modified" is selected – Ben Grossmann Oct 08 '24 at 21:07
  • @BenGrossmann Thank you. I do not know when and how I changed it from "Highest score" (which is the default) to "date modified". I guess, I am still not comfortable with the UI of this site. I'm using a dark theme on my browser which makes it difficult to see the "Sorted by". – Nothing special Oct 08 '24 at 21:12

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