Let $\mathbb{F}_{p}$ be the set of (residue classes of) integers $\bmod p$, and let

$$G=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right): a, b, c, d \in \mathbb{F}_{p}, a d-b c \neq 0\right\}$$

Show that $G$ is a group under multiplication. [You may assume throughout this question that multiplication of matrices is associative.]

Let $X$ be the set of 2-dimensional column vectors with entries in $\mathbb{F}_{p}$. Show that the mapping $G \times X \rightarrow X$ given by

$$\left(\left(\begin{array}{ll} a & b \\ c & d \end{array}\right),\left(\begin{array}{l} x \\ y \end{array}\right)\right) \mapsto\left(\begin{array}{l} a x+b y \\ c x+d y \end{array}\right)$$

is a group action.

Let $g \in G$ be an element of order $p$. Use the orbit-stabilizer theorem to show that there exist $x, y \in \mathbb{F}_{p}$, not both zero, with

$$g\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{l} x \\ y \end{array}\right)$$

Deduce that $g$ is conjugate in $G$ to the matrix

$$\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)$$