Group factorisation theorem
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Contents
Statement
 If [ilmath]N\subseteq\text{Ker}(\varphi)[/ilmath] then [ilmath]\varphi[/ilmath] factors uniquely through [ilmath]\pi[/ilmath] to yield [ilmath]\bar{\varphi}:G/N\rightarrow H[/ilmath] given by [ilmath]\bar{\varphi}:[g]\mapsto\varphi(g)[/ilmath] ^{[Note 2]} and [ilmath]\bar{\varphi} [/ilmath] is a group homomorphism.
Additionally we have [ilmath]\varphi=\overline{\varphi}\circ\pi[/ilmath] (or in other terms, the diagram on the right commutes)
Proof
The bulk of this proof involves invoking the function factorisation theorem:
Let [ilmath]f:X\rightarrow Y[/ilmath] and [ilmath]w:X\rightarrow W[/ilmath] be functions.

Applying the function factorisation theorem
Before we can apply this theorem we need to check that the objects in play are eligible (satisfy the requirements to factor) for the theorem. We set up as follows:
 Let [ilmath]G[/ilmath], [ilmath]H[/ilmath] be groups, let [ilmath]N\trianglelefteq G[/ilmath] and let [ilmath]\varphi:G\rightarrow H[/ilmath] be a group homomorphism with [ilmath]N\subseteq\text{Ker}(\varphi)[/ilmath], then:
 [ilmath]\forall a,b\in X\big[ [\pi(a)=\pi(b)]\implies[\varphi(a)=\varphi(b)]\big][/ilmath]
 where [ilmath]\pi:G\rightarrow G/N[/ilmath] is the canonical projection of the quotient group
Proof
Let [ilmath]a,b\in X[/ilmath] be given
 If [ilmath]\pi(a)\ne\pi(b)[/ilmath] then it doesn't matter what [ilmath]\varphi(a)[/ilmath] and [ilmath]\varphi(b)[/ilmath] are, as for logical implication if the LHS is false, it doesn't matter what the RHS for the implication to be considered true.
 If [ilmath]\pi(a)=\pi(b)[/ilmath] then we require [ilmath]\varphi(a)=\varphi(b)[/ilmath] for the implication to be true.
 By hypothesis we have [ilmath]\pi(a)=\pi(b)[/ilmath] so:
 [ilmath]\implies\pi(a)[\pi(b)]^{1}=e[/ilmath] (here [ilmath]e[/ilmath] will denote the identity of whatever group makes sense in the context, in this case [ilmath]G/N[/ilmath] and [ilmath]\implies[/ilmath] may not be the strongest logical relation that can be used)
 [ilmath]\implies\pi(a)\pi(b^{1})=e[/ilmath]
 [ilmath]\implies\pi(ab^{1})=e[/ilmath]
 [ilmath]\implies ab^{1}\in\text{Ker}(\pi)[/ilmath] But! The kernel of the canonical projection of a quotient group is the normal subgroup of the quotient
 [ilmath]\implies ab^{1}\in N[/ilmath]
 [ilmath]\implies ab^{1}\in\text{Ker}(\varphi)[/ilmath] (by hypothesis, as [ilmath]N\subseteq\text{Ker}(\varphi)[/ilmath] using the impliessubset relation)
 [ilmath]\implies \varphi(ab^{1})=e[/ilmath] ([ilmath]e[/ilmath] is the identity of [ilmath]H[/ilmath] this time)
 [ilmath]\implies \varphi(a)[\varphi(b)]^{1}=e[/ilmath]
 [ilmath]\implies \varphi(a)=\varphi(b)[/ilmath]
 We have shown [ilmath]\pi(a)=\pi(b)\implies\varphi(a)=\varphi(b)[/ilmath] as required.
 By hypothesis we have [ilmath]\pi(a)=\pi(b)[/ilmath] so:
Result
We have shown that the factor (function) theorem is eligible for application here, thus we apply it and yield:
 A function [ilmath]\bar{\varphi}:G/N\rightarrow H[/ilmath] given by [ilmath]\bar{\varphi}:[u]\rightarrow\varphi(u)[/ilmath]^{[Note 3]}^{[Note 4]}
 By the surjectivity of the canonical project of the quotient group ([ilmath]\pi:G\rightarrow G/N[/ilmath]) we know from the factor theorem that the induced function, [ilmath]\bar{\varphi}:G/N\rightarrow H[/ilmath] is the unique function that satisfies:
 [ilmath]\varphi=\bar{\varphi}\circ\pi[/ilmath] (there is no other [ilmath]\bar{\varphi}'[/ilmath] with this property that is distinct from [ilmath]\bar{\varphi} [/ilmath])
Showing [ilmath]\bar{\varphi} [/ilmath] is a group homomorphism
We now have a function:
 [ilmath]\bar\varphi:G/N\rightarrow H[/ilmath] given by [ilmath]\bar\varphi:[u]\mapsto\varphi(u)[/ilmath]. We claim this is a group homomorphism. To show this we must show that:
 [ilmath]\forall a,b\in G/N[\bar{\varphi}(a,b)=\bar{\varphi}(a)\bar{\varphi}(b)][/ilmath]
We shall write [ilmath][u]\in G/N[/ilmath] and choose [ilmath]u\in[u][/ilmath] to represent the classes in what follows, however as has been mentioned in the notes several times, this isn't a problem.
 Let [ilmath][u],[v]\in G/N[/ilmath] be given; as usual [ilmath][uv]=[u]*[v][/ilmath] where [ilmath]*[/ilmath] is the binary operation of the [ilmath]G/N[/ilmath] group.
 Then [ilmath]\bar{\varphi}([uv])=\varphi(uv)=\varphi(u)\varphi(v)[/ilmath] as [ilmath]\varphi[/ilmath] is a group homomorphism.
 Note that [ilmath]\bar{\varphi}([u])\bar{\varphi}([v])=\varphi(u)\varphi(v)[/ilmath]
 So [ilmath]\varphi(u)\varphi(v)=\bar{\varphi}([u])\bar{\varphi}([v])=\bar{\varphi}([uv])[/ilmath]
 We have shown [ilmath]\bar{\varphi}([u])\bar{\varphi}([v])=\bar{\varphi}([uv])[/ilmath] as required.
Thus [ilmath]\bar{\varphi} [/ilmath] is a grop homomorphism from [ilmath]G/N[/ilmath] to [ilmath]H[/ilmath]
See also
Notes
 ↑ The notation [ilmath]A\trianglelefteq B[/ilmath] means [ilmath]A[/ilmath] is a normal subgroup of the group [ilmath]B[/ilmath].
 ↑ This may look strange as obviously you're thinking "what if we took a different representative [ilmath]h\in[g][/ilmath] with [ilmath]h\ne g[/ilmath], then we'd have [ilmath]\varphi(h)[/ilmath] instead of [ilmath]\varphi(g)[/ilmath]!", these are actually the same, see Factor (function) for more details, I shall explain this here.
 Technically we have this: [ilmath]\bar{\varphi}:u\mapsto\varphi(\pi^{1}(u))[/ilmath] for the definition of [ilmath]\bar{\varphi} [/ilmath]
 Note though that if [ilmath]g,h\in\pi^{1}(u)[/ilmath] that:
 [ilmath]\pi(g)=\pi(h)=u[/ilmath] and by hypothesis we have [ilmath][\pi(x)=\pi(y)]\implies[\varphi(x)=\varphi(y)][/ilmath]
 Thus [ilmath]\varphi(g)=\varphi(h)[/ilmath]
 [ilmath]\pi(g)=\pi(h)=u[/ilmath] and by hypothesis we have [ilmath][\pi(x)=\pi(y)]\implies[\varphi(x)=\varphi(y)][/ilmath]
 So whichever representative of [ilmath][g][/ilmath] we use [ilmath]\varphi(h)[/ilmath] for [ilmath]h\in[g][/ilmath] is the same.
 Note though that if [ilmath]g,h\in\pi^{1}(u)[/ilmath] that:
 This is actually all dealt with as a part of factor (function) not this theorem. However it is worth illustrating.
 Technically we have this: [ilmath]\bar{\varphi}:u\mapsto\varphi(\pi^{1}(u))[/ilmath] for the definition of [ilmath]\bar{\varphi} [/ilmath]
 ↑ This is just a nicer way of writing:
 [ilmath]\bar{\varphi}:g\mapsto \varphi(\pi^{1}(g))[/ilmath]
 ↑ Please note, again, this may be an abuse of notation, but the result is well defined, as for [ilmath]a,b\in\pi^{1}(x)[/ilmath] we see [ilmath]\varphi(a)=\varphi(b)[/ilmath] by hypothesis.
References
