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 $\left(\begin{array}{c c}a &amp; b \\ c &amp; d\end{array}\right)$ 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 $\left(\begin{array}{c c}a &amp; b \\ c &amp; d\end{array}\right)$ where $ad-bc=0$ (mod 3). That means there are at most $81-44 = 37$ such $\left(\begin{array}{c c}a &amp; b \\ c &amp; d\end{array}\right)$ 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, $|G|=(p^2-1)(p^2-p)$.
Let G be group of all 2 cross 2 matrices(a b c d){where in 1st row has a and b,2nd row has c and d}and a,b,c,d are integers modulo 3,relative to matrix multiplication.show ghat o(G)=48.(topics in algebra by i n hetstein with 2nd edition.capter-2 problem number 25)
Solution:-According to the question this is a general linear group of integers modulo n.for this in general we have a formula to find the order of group.
o(GL(n z_p))=(p^n-1)(p^n-p)...(p^n-p^n-1)
So according to this formula here n=2 and p=3.Now put these value in the above formula we will get order of general linear equal to 48.