> [!NOTE] **Definition** (Image of a group homomorphism) > Let $G$ and $H$ be [[Groups|groups]] and let $\phi: G \to H$ be a [[Homomorphisms of groups|homomorphism]]. Then the image of $\phi$ is defined as the [[Image of a set under a function|image]] of $G$ under $\phi,$ denoted $\text{Im }\phi = \{ \phi(g) \mid g \in G \} $is called the *image* of $\phi$. # Properties Note [[Image of Homomorphism of Groups is Subgroup of Codomain]].