Difference between revisions of "First group isomorphism theorem"
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| − | ==[[First group isomorphism theorem | + | ==[[First group isomorphism theorem/Statement|Statement]]== |
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Revision as of 19:36, 15 July 2016
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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
Contents
| 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 | |
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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]
Proof
Notes
References
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