> [!NOTE] **Definition** (Group Homomorphism) > > Let $(G, \diamond)$ and $(H, *)$ be [[Group|groups]] We say that the [[Function|mapping]] $\phi: G \to H$ is a *homomorphism* if it satisfies $\phi(g_{1} \diamond g_{2}) = \phi(g_{1}) * \phi(g_{2})$for all $g_{1}, g_{2} \in G$. > [!Example] Example > Let $G= \mathbb{Z}$ and let $H=\mathbb{Z}\text{/} n\mathbb{Z}$. Define a function $\phi: G \to H$ as follows $\phi(m)=[m]_{n}$Now $\phi(k)+\phi(l) = [k]_{n} +_{n} [l]_{n} = [k+l]_{n} = \phi(k+l)$. So $\phi$ is a homomorphism. > Ker $\phi = \{ k \in Z \mid [k]_{n}= [0]_{n} \} =n \mathbb{Z}$. > Im $\phi= \{ \phi(k) \mid k \in \mathbb{Z} \} = \{ [k]_{n} \mid k \in \mathbb{Z} \}= \mathbb{Z} \text{/}n\mathbb{Z}$ # Properties Note that a [[Homomorphism of Groups Preserves Identity|homomorphism fixes the identity element]]. **Kernel & Image**: The [[Image of Homomorphism of Groups|image]] of a homomorphism is the image of its domain under the homomorphism and its [[Kernel of Homomorphism of Groups|kernel]] is the pre-image of the identity of its codomain. Note that the kernel and image of $\phi$ are subgroups of $G$ and $H$ respectively. A [[Homomorphism of Groups is Injective iff its Kernel only Contains the Identity of Its Domain|homomorphism is injective iff the kernel is trivial]]. **Types**: If $\phi$ is a bijection then its is called a [[Isomorphism of Groups|group isomorphism]].