Difference between revisions of "First group isomorphism theorem"
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{{:First group isomorphism theorem/Statement}} | {{:First group isomorphism theorem/Statement}} | ||
==Proof== | ==Proof== | ||
| + | * See [[Notes:Proof of the first group isomorphism theorem]] | ||
==Notes== | ==Notes== | ||
Revision as of 16:40, 16 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
[hide]| First isomorphism theorem | |
Where θ is an isomorphism. | |
| Properties | |
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| something |
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
Proof
Notes
References
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