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 | ||
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{{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>< | + | |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 | ||
| − | }} | + | }}__TOC__ |
==[[First group isomorphism theorem/Statement|Statement]]== | ==[[First group isomorphism theorem/Statement|Statement]]== | ||
{{:First group isomorphism theorem/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== | ==Proof== | ||
| + | * 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:
- 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 | |
| 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
- 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
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