What does it mean to say that a subgroup $K$ of a group $G$ is normal?

Let $\phi: G \rightarrow H$ be a group homomorphism. Is the kernel of $\phi$ always a subgroup of $G$ ? Is it always a normal subgroup? Is the image of $\phi$ always a subgroup of $H$ ? Is it always a normal subgroup? Justify your answers.

Let $\mathrm{SL}(2, \mathbb{Z})$ denote the set of $2 \times 2$ matrices $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a, b, c, d \in \mathbb{Z}$ and $a d-b c=1$. Show that $\mathrm{SL}(2, \mathbb{Z})$ is a group under matrix multiplication. Similarly, when $\mathbb{Z}_{2}$ denotes the integers modulo 2 , let $\mathrm{SL}\left(2, \mathbb{Z}_{2}\right)$ denote the set of $2 \times 2$ matrices $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ with $a, b, c, d \in \mathbb{Z}_{2}$ and $a d-b c=1$. Show that $\mathrm{SL}\left(2, \mathbb{Z}_{2}\right)$ is also a group under matrix multiplication.

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}_{2}$ send each integer to its residue modulo 2 . Show that

$$\phi: \mathrm{SL}(2, \mathbb{Z}) \rightarrow \mathrm{SL}\left(2, \mathbb{Z}_{2}\right) ; \quad\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \mapsto\left(\begin{array}{ll} f(a) & f(b) \\ f(c) & f(d) \end{array}\right)$$

is a group homomorphism. Show that the image of $\phi$ is isomorphic to a permutation group.