First group isomorphism theorem/Statement

Statement

Let $(G,*)$ and $(H,*)$ be groups. Let $\varphi:G\rightarrow H$ be a group homomorphism, then:

• $G/\text{Ker}(\varphi)\cong\text{Im}(\varphi)$
  • Explicitly we may state this as: there exists a group isomorphism between $G/\text{Ker}(\varphi)$ and $\text{Im}(\varphi)$.

Note: the special case of $\varphi$ being surjective, then $\text{Im}(\varphi)=H$, so we see $G/\text{Ker}(\varphi)\cong H$