View source for 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
{{Infobox
|title=<span style="font-size:0.85em;">First isomorphism theorem</span>
|above=<div style="overflow:hidden;"><span style="font-size:1.3em;">{{M|1=\begin{xy}\xymatrix{A \ar[r]^\varphi \ar[d]_{\pi} & B \\ A/\text{Ker}(\varphi) \ar@{.>}[r]^-{\theta}& \text{Im}(\varphi) \ar@{^{(}->}[u]^i }\end{xy} }}</span></div>Where {{M|\theta}} is an [[group isomorphism|isomorphism]].
|header1=Properties
|data1=something
}}__TOC__
==[[First group isomorphism theorem/Statement|Statement]]==
{{:First group isomorphism theorem/Statement}}
==Useful [[corollary|corollaries]]==
# [[An injective group homomorphism means the group is isomorphic to its image]]
#* If {{M|\varphi:A\rightarrow B}} is an ''[[injective]]'' [[group homomorphism]] then {{M|A\cong \text{Im}(\varphi)}}
# [[A surjective group homomorphism means the target is isomorphic to the quotient of the domain and the kernel]]
#* If {{M|\varphi:A\rightarrow B}} is a ''[[surjective]]'' [[group homomorphism]] then {{M|A/\text{Ker}(\varphi)\cong B}}
==Proof==
* See [[Notes:Proof of the first group isomorphism theorem]]

==Notes==
<references group="Note"/>
==References==
<references/>
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{{Theorem Of|Abstract Algebra|Group Theory}}