A group homomorphism between two groups $G$ and $H$ is a function $f : G \to H$ such that for all $a$, $b \in G$,
\[f(ab) = f(a) f(b).\]
The kernel of a group homomorphism $f : G \to H$ is
\[\ker f = f^{-1}[\{i\}],\]where $i$ is the neutral element of $H$.
Let $G$ be a group with neutral element $e$ and $H$ be a group with neutral element $i$. For any group homomorphism $f : G \to H$:
- \[f(e) = i.\]
For all $a\in G$,
\[f(a^{-1}) = f(a)^{-1}.\]
For any group homomorphism $f : G \to H$, $\ker f$ is a normal subgroup of $G$.
For any group homomorphism $f : G \to H$, $\im f$ is a subgroup of $H$.
Let $f : G \to H$ be a group homomorphism. For all $a$, $b \in G$,
\[f(a) = f(b) \enspace\lrimp\enspace ab^{-1} \in \ker f.\]
As a corollary:
Let $f:G\to H$ be a group homomorphism and $e$ be the neutral element of $G$. $f$ is a monomorphism if and only if
\[\ker f = \{e\}.\]