Order of general linear group of $2 \times 2$ matrices over $\mathbb{Z}_3$
From problem 2.3.25 in Topics in Algebra, 2$\varepsilon$ by I. N. Herstein:
Let $G$ be the group of all $2 \times 2$ matrices $\left(\begin{array}{c c}a & b \\ c & d\end{array}\right)$ where $ad-bc \ne 0$ and $a,b,c,d$ are integers modulo 3, relative to matrix multiplication. Show that $o(G) = 48$.
I know that $o(G) \le 3^4 = 81$, since $a,b,c,d$ can each take one of 3 values (mod 3). I attempted to tighten this bound by finding the number of matrices such that $ad=bc$ (mod 3):
- Suppose $ad=bc=0$ (mod 3). Then ($a = 0$ or $d = 0$) and ($b = 0$ or $c = 0$), leading to 36 possible values for $(a,b,c,d)$.
- Suppose $ad=bc=1$ (mod 3). Then ($a=d=1$ or $a=d=2$) and ($b=c=1$ or $b=c=2$), leading to 4 possible values for $(a,b,c,d)$.
- Suppose $ad=bc=2$ (mod 3). Then ($(a,d)=(1,2)$ or $(a,d)=(2,1)$) and ($(b,c)=(1,2)$ or $(b,c)=(2,1)$), leading to 4 possible values for $(a,b,c,d)$.
So, there are in total $36+4+4 = 44$ such $\left(\begin{array}{c c}a & b \\ c & d\end{array}\right)$ where $ad-bc=0$ (mod 3). That means there are at most $81-44 = 37$ such $\left(\begin{array}{c c}a & b \\ c & d\end{array}\right)$ where $ad-bc\ne 0$, i.e., $o(G) \le 37$. However, this contradicts the problem. Where did I go wrong? Can someone set me on the right path?
Vector $a\choose c$ must be non-zero, this allows $p^2-1$ choices. Vector $b\choose d$ must not be a multiple of $a\choose c$, this allows $p^2-p$ choices. Thus, $|G|=(p^2-1)(p^2-p)$.
Use the following result: A matrix is non-singular $\iff$ the columns are linearly indipendent.
Hints:
(1) In how many ways can you choose the first column for a matrix in $\,G\,$ ?
(2) Now, in how many ways can you choose the second column?
If you know the vector field $\,\Bbb F_3^2=\left(\Bbb Z/3\Bbb Z\right)^2\,$ then $\,G\,$ is the set of all the invertible matrices over this vector space, and the hints above basically ask: how many different (ordered, of course) basis are there for $\,\Bbb F_3^2\,$ over $\,\Bbb F_3\,$ ?
There are only $25$ matrices with $ad=bc=0$. To get $ad=0$ you must have $a=0$ ($3$ cases) or $b=0$ ($3$ cases), but two of those $6$ cases are identical ($a=b=0$), so you really have only $5$. After this correction you’re throwing out $25+4+4=33$ matrices, leaving the desired $48$.