# First group isomorphism theorem/Statement

From Maths

## Statement

Let [ilmath](G,*)[/ilmath] and [ilmath](H,*)[/ilmath] be groups. Let [ilmath]\varphi:G\rightarrow H[/ilmath] be a group homomorphism, then^{[1]}:

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

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