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= }}</span>< | + | |above=<div style="overflow:hidden;"><span style="font-size:1.3em;">{{M|1= }}</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]] | * See [[Notes:Proof of the first group isomorphism theorem]] | ||
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 θ is an isomorphism. | |
| Properties | |
|---|---|
| something |
Contents
[hide]Statement
Let (G,∗) and (H,∗) be groups. Let φ:G→H be a group homomorphism, then[1]:
- G/Ker(φ)≅Im(φ)
- Explicitly we may state this as: there exists a group isomorphism between G/Ker(φ) and Im(φ).
Note: the special case of φ being surjective, then Im(φ)=H, so we see G/Ker(φ)≅H
Useful corollaries
- An injective group homomorphism means the group is isomorphic to its image
- If φ:A→B is an injective group homomorphism then A≅Im(φ)
- A surjective group homomorphism means the target is isomorphic to the quotient of the domain and the kernel
- If φ:A→B is a surjective group homomorphism then A/Ker(φ)≅B
Proof
Notes
References
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