Difference between revisions of "Notes:Proof of the first group isomorphism theorem"
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{| class="wikitable" border="1" style="overflow:hidden;" | {| class="wikitable" border="1" style="overflow:hidden;" | ||
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− | | style="font-size:1.2em;" | <center><m>\xymatrix{ G \ar@{-->}[dr] | + | | style="font-size:1.2em;" | <center><m>\xymatrix{ G \ar@{-->}[dr]_(.17){\varphi'} \ar[d]_\pi \ar[r]^\varphi & H \\ G/\text{Ker}(\varphi) \ar@{.>}[ur]_(.8){\bar{\varphi} } \ar@{.>}[r]_-\theta & \text{Im}(\varphi) \ar@{^{(}->}[u]_i }</m></center> |
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! Diagram of morphisms in play | ! Diagram of morphisms in play | ||
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** Which is of course a group homomorphism. | ** Which is of course a group homomorphism. | ||
** And has the property: {{M|1=\varphi=\bar{\varphi}\circ\pi}} | ** And has the property: {{M|1=\varphi=\bar{\varphi}\circ\pi}} | ||
+ | * Additionally, I take it as trivial that: | ||
+ | ** {{M|1=\varphi=i\circ\varphi'}} | ||
+ | ===Proof=== | ||
+ | * Note that {{M|1=\varphi=i\circ\varphi'}} and {{M|1=\varphi'=\theta\circ\pi}} - by substitution we see: | ||
+ | ** {{M|1=\varphi=i\circ\theta\circ\pi }} | ||
+ | This shows that the diagram commutes, we only need to show that {{M|\theta}} is a [[group isomorphism]] to finish the proof. | ||
+ | * I would like to do something like {{M|1=\varphi=\bar{\varphi}\circ\theta^{-1}\circ\varphi'}} but I can't as {{M|\theta^{-1} }} might not be a function. | ||
+ | Lets try the "brute force" approach of just showing it. | ||
+ | # {{M|\theta}} is [[surjective]]. | ||
+ | #* While I have failed (I think... it was a few days ago and I wasn't pleased with it) to do it, I think the following is promising: | ||
+ | #** Suppose {{M|\theta}} is not surjective, then we cannot have {{M|1=\varphi'=\theta\circ\pi}} (as {{M|\varphi'}} is surjective) | ||
+ | # {{M|\theta}} is [[injective]] | ||
+ | #* Suppose {{M|1=\theta(x)=\theta(y)}}, we wish to show that this means {{M|1=x=y}} | ||
+ | #** The gist is this: {{M|1=\theta([u])=\theta([v])\implies\varphi(u)=\varphi(v)\implies\varphi(uv^{-1})=e}} thus {{M|uv^{-1}\in\text{Ker}(\varphi)}} | ||
+ | #*** So {{M|1=\pi(uv^{-1})\in\text{Ker}(\varphi)}} so {{M|1=\pi(uv^{-1})=[e]}} (the coset that is the normal subgroup {{M|\text{Ker}(\varphi)}} itself) | ||
+ | #*** Thus {{M|1=\pi(uv^{-1})=[e]\implies\pi(u)=[e]\pi(v)\implies[u]=[v]}} | ||
+ | #** We have shown {{M|1=\theta([u])=\theta([v])\implies[u]=[v]}} | ||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
{{Notes|Abstract Algebra|Group Theory}} | {{Notes|Abstract Algebra|Group Theory}} |
Latest revision as of 17:41, 16 July 2016
Claim
Let G and H be groups, let φ:G→H be any group homomorphism, then:
- G/Ker(φ)≅Im(φ)
Or, alternatively:
- There exists a group isomorphism, θ:G/Ker(φ)→Im(φ) such that the following diagram commutes:
- (so φ=i∘θ∘π) where i:Im(φ)→H is the canonical injection, i:h↦h. It is a group homomorphism.
Proof
First note:
- We get a function, φ′:G→Im(φ) I'll call the "canonical surjection", given by φ′:g↦φ(g).
- We can factor φ′ through π (using the group factorisation theorem) to get θ:G/Ker(φ)→Im(φ)
- Which is of course a group homomorphism.
- And has the property: φ′=θ∘π
- We can factor φ through π to, to give ˉφ:G/Ker(φ)→H
- Which is of course a group homomorphism.
- And has the property: φ=ˉφ∘π
- Additionally, I take it as trivial that:
- φ=i∘φ′
Proof
- Note that φ=i∘φ′ and φ′=θ∘π - by substitution we see:
- φ=i∘θ∘π
This shows that the diagram commutes, we only need to show that θ is a group isomorphism to finish the proof.
- I would like to do something like φ=ˉφ∘θ−1∘φ′ but I can't as θ−1 might not be a function.
Lets try the "brute force" approach of just showing it.
- θ is surjective.
- While I have failed (I think... it was a few days ago and I wasn't pleased with it) to do it, I think the following is promising:
- Suppose θ is not surjective, then we cannot have φ′=θ∘π (as φ′ is surjective)
- While I have failed (I think... it was a few days ago and I wasn't pleased with it) to do it, I think the following is promising:
- θ is injective
- Suppose θ(x)=θ(y), we wish to show that this means x=y
- The gist is this: θ([u])=θ([v])⟹φ(u)=φ(v)⟹φ(uv−1)=e thus uv−1∈Ker(φ)
- So π(uv−1)∈Ker(φ) so π(uv−1)=[e] (the coset that is the normal subgroup Ker(φ) itself)
- Thus π(uv−1)=[e]⟹π(u)=[e]π(v)⟹[u]=[v]
- We have shown θ([u])=θ([v])⟹[u]=[v]
- The gist is this: θ([u])=θ([v])⟹φ(u)=φ(v)⟹φ(uv−1)=e thus uv−1∈Ker(φ)
- Suppose θ(x)=θ(y), we wish to show that this means x=y