Let $K$ be a subgroup of a group $G$. Prove that $K$ is normal if and only if there is a group $H$ and a homomorphism $\phi: G \rightarrow H$ such that

$$K=\{g \in G: \phi(g)=1\}$$

Let $G$ be the group of all $2 \times 2$ matrices $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a, b, c, d$ in $\mathbb{Z}$ and $a d-b c=1$. Let $p$ be a prime number, and take $K$ to be the subset of $G$ consisting of all $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a \equiv d \equiv 1(\bmod p)$ and $c \equiv b \equiv 0(\bmod p) .$ Prove that $K$ is a normal subgroup of $G .$