Problem from Herstein (Group Theory)

This is the problem from Topics in Algebra by I. N. Herstein. Part of Example No. 2.2.9:

Let $G$ be the set of all $2 \times 2$ matrices $\left( {\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right)$ where $a, b, c, d$ are integers modulo $p$, ($p$ a prime number), such that $ad-bc \neq 0$. Under matrix multiplication, prove that $G$ is a non-abelian finite group for any general $p$. All the multiplication and additions of the entries are those with modulo $p$.

Till now, I am solving it for small values of $p$, writing down elements explicitly as suggested in the example itself. I would like to know proof for any general $p$.

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What part of the request are you not able to prove in the general case? Is proving G a group, proving it non abelian or showing it finite that bother you? –  Marco Vergura Jul 6 at 16:30
Are you familiar with the language and concepts of vector spaces over the field $\Bbb{Z}/p\Bbb{Z}$? This makes a huge difference here. –  Jyrki Lahtonen Jul 6 at 16:36
I would like to know the whole solution... (I did things in bits and pieces, like it being in group etc.) No, I am not familiar with language you mentioned. I have done some courses in basic linear algebra, calculus, mathematical physics and have just started doing group theory. –  Amit Seta Jul 6 at 16:40
You mean $ad-bc\not\equiv 0(\mod p)$ right? –  Gina Jul 6 at 16:46
Sorry, yes Gina I should have mentioned that. –  Amit Seta Jul 6 at 16:48

For general $p$ prime, the group in question is $GL_{2}(\mathbb{Z}/p)$, the general linear group of degree $2$ over the field $\mathbb{Z}/p$, the ring of integers modulo $p$. For a matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $ad-bc = \text{det}(A)$. For two matrices $A,B \in G$, we have that $\text{det}(AB)=\text{det}(A)\text{det}(B) \neq 0$, because the unit group of $\mathbb{Z}/p$ consists of all non-zero elements, i.e $\text{det}:G \rightarrow \mathbb{Z}/p^{\times}$ is a group homorphism. So $G$ is closed under matrix multiplication. The normal formula for the inverse of a $2$x$2$ matrix holds, the identity element is $I_{2}$, and associativity holds because it holds in $\mathbb{Z}/p$. So $G=GL_{2}(\mathbb{Z}/p)$ is a group.

$G$ is nonabelian for any $p$ prime as $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$, but $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$

The finiteness of $G$ is clear because the total number of possible $2$x$2$ matrices over $\mathbb{Z}/p$ is $p^{4}$, and $G$ is a subset of this. $\square$

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The examples is given in the book before speaking anything about homorphism, linear groups, rings and fields. –  Amit Seta Jul 6 at 19:00
Are any of those topics covered in the book later? If not, just note that if $\text{det}(A), \text{det}(B)$ are $\neq 0$ mod $p$, then $\text{det}(A)\text{det}(B)$ is also $\neq 0$ mod $p$, otherwise $p$ divides $\text{det}(A)\text{det}(B)$ which implies $\text{det}(A)$ or $\text{det}(B)$ are $0$ mod $p$, a contradiction. –  Edward ffitch Jul 6 at 19:06
Yes, they are actually covered later. But since, the question is given before that I assume the knowledge of those is not mandatory. The solution you provided now is very similar to another answer. Thanks –  Amit Seta Jul 6 at 19:09
Your assumption is correct, and you are welcome. =) –  Edward ffitch Jul 6 at 19:11

For an elementary method that simply involve multiplying out everything...

To check for closure, simply multiplying out everything.

Associative can once again be checked by multiplying out everything.

The identity is $\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}$

The inverse is $(ad-bc)^{p-2}\begin{pmatrix}d & -b\\-c & a\end{pmatrix}$ (get this from normal rule for 2x2 matrix inverse, and the inverse of determinant is just from FLT).

To prove that it is nonabelian, consider $\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}$ and $\begin{pmatrix}1 & 0\\1 & 1\end{pmatrix}$ which is guaranteed to work for any $p$.

Finiteness is from the fact that each component is finite.

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lemma: if $gcd(a,n)=1$ then $a^{-1}$ exist in $\mathbb Z_n$.

Let $A\in G$ then $A^{-1}=(det(A))^{-1}Adj(A)$, In that case, we need to check whether $det(A)^{-1}$ exist in $\mathbb Z_p$.

Since $p\nmid det(A)$ then $gcd(p,det(A))=1$.

Since $det(AB)=det(A)det(B)$ which shows that if $A,B\in G$ then $AB\in G$.

We are done.

Finiteness is obvious as number of the all possible matrices is $p^4$ so this number is less than $p^4$.

I left you to show that it is nonabelian.

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