G is a group given by :
G = { q | ad - bc = 1} where a, b, c, d are integers modulo p, a prime number and
Find the o(G)
let $\displaystyle L$ be the group of $\displaystyle 2 \times 2$ invertible matrices with entries in $\displaystyle \mathbb{F}_p.$ define the homomorphism $\displaystyle f: L \longrightarrow \mathbb{F}_p^{\times}$ by $\displaystyle f(A)=\det A.$ then $\displaystyle G=\ker f$ and $\displaystyle L/G \cong \mathbb{F}_p^{\times}.$ thus $\displaystyle |G|=\frac{|L|}{p-1}.$
to find $\displaystyle |L|$ look at the elements of $\displaystyle L$ as invertible transformations from $\displaystyle V$ to $\displaystyle V,$ where $\displaystyle V$ is a vector space of dimension $\displaystyle 2$ over $\displaystyle \mathbb{F}_p.$ let $\displaystyle \{v_1,v_2 \}$ be a basis for $\displaystyle V.$ let $\displaystyle T \in L.$
then $\displaystyle T(v_1)$ may be defined to be any non-zero element of $\displaystyle V.$ so $\displaystyle T(v_1)$ has $\displaystyle p^2 - 1$ options. now $\displaystyle T(v_2)$ can be anything in $\displaystyle V - <T(v_1)>.$ so there are $\displaystyle p^2-p$ options for $\displaystyle T(v_2).$
hence $\displaystyle |L|=(p^2-1)(p^2-p)$ and therefore $\displaystyle |G|=\frac{|L|}{p-1}=p(p^2-1). \ \ \Box$
the above method can be used to, in general, find the order of the group of all $\displaystyle n \times n$ matrices with determinant 1 and with entries from some finite field. (just copy the above proof!)