Difference between revisions of "First group isomorphism theorem"

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Overview of the group isomorphism theorems - all 3 theorems in one place
Overview of the isomorphism theorems - the first, second and third are pretty much the same just differing by what objects they apply to

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

G/Ker(φ)≅Im(φ)
- 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 $G/\text{Ker}(\varphi)\cong H$

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-==[[First group isomorphism theorem\Statement|Statement]]==
+==[[First group isomorphism theorem/Statement|Statement]]==
-{{:First group isomorphism theorem\Statement}}
+{{:First group isomorphism theorem/Statement}}
 ==Proof==

Properties
First isomorphism theorem
Where $\theta$ is an isomorphism.
something