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 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
 
:* [[Overview of the isomorphism theorems]] - the first, second and third are pretty much the same just differing by what objects they apply to
__TOC__
 
 
{{Infobox
 
{{Infobox
 
|title=<span style="font-size:0.85em;">First isomorphism theorem</span>
 
|title=<span style="font-size:0.85em;">First isomorphism theorem</span>
|above=<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><br/>Where {{M|\theta}} is an [[group isomorphism|isomorphism]].
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|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
 
|header1=Properties
 
|data1=something
 
|data1=something
}}
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}}__TOC__
 
==[[First group isomorphism theorem/Statement|Statement]]==
 
==[[First group isomorphism theorem/Statement|Statement]]==
 
{{:First group isomorphism theorem/Statement}}
 
{{:First group isomorphism theorem/Statement}}
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==Useful [[corollary|corollaries]]==
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# [[An injective group homomorphism means the group is isomorphic to its image]]
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#* If {{M|\varphi:A\rightarrow B}} is an ''[[injective]]'' [[group homomorphism]] then {{M|A\cong \text{Im}(\varphi)}}
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# [[A surjective group homomorphism means the target is isomorphic to the quotient of the domain and the kernel]]
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#* If {{M|\varphi:A\rightarrow B}} is a ''[[surjective]]'' [[group homomorphism]] then {{M|A/\text{Ker}(\varphi)\cong B}}
 
==Proof==
 
==Proof==
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* See [[Notes:Proof of the first group isomorphism theorem]]
  
 
==Notes==
 
==Notes==

Latest revision as of 04:17, 20 July 2016

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Saving work
Note:
First isomorphism theorem
[ilmath]\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}[/ilmath]
Where [ilmath]\theta[/ilmath] is an isomorphism.
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

  1. An injective group homomorphism means the group is isomorphic to its image
  2. A surjective group homomorphism means the target is isomorphic to the quotient of the domain and the kernel

Proof

Notes

References

  1. Abstract Algebra - Pierre Antoine Grillet