Group Homomorphism

Definition 1. A group homomorphism is a magma homomorphism between groups.

Proposition 1. Let $G$ be a group with neutral element $e$ and $H$ be a group with neutral element $i$. For any group homomorphism $f : G \to H$:

1. (Neutral-preserving)

   \[f(e) = i.\]

2. (Inverse-preserving) For all $a\in G$,

   \[f(a^{-1}) = f(a)^{-1}.\]

Proposition 2. For any group homomorphism $f : G \to H$, $\ker f$ is a normal subgroup of $G$.

Proposition 3. For any group homomorphism $f : G \to H$, $\im f$ is a subgroup of $H$.

Proposition 4. Let $f : G \to H$ be a group homomorphism. For all $a$, $b \in G$,

\[f(a) = f(b) \enspace\lrimp\enspace ab^{-1} \in \ker f.\]

Proposition 5. 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

\[\ker f = \{e\}.\]