First group isomorphism theorem
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 Note:
 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
First isomorphism theorem  
isomorphism.  Where [ilmath]\theta[/ilmath] is an
Properties  

something 
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]
Useful corollaries
 An injective group homomorphism means the group is isomorphic to its image
 If [ilmath]\varphi:A\rightarrow B[/ilmath] is an injective group homomorphism then [ilmath]A\cong \text{Im}(\varphi)[/ilmath]
 A surjective group homomorphism means the target is isomorphic to the quotient of the domain and the kernel
 If [ilmath]\varphi:A\rightarrow B[/ilmath] is a surjective group homomorphism then [ilmath]A/\text{Ker}(\varphi)\cong B[/ilmath]
Proof
Notes
References
