Difference between revisions of "Group factorisation theorem"

Revision as of 16:39, 16 July 2016 (view source) Alec (Talk | contribs) m ← Older edit		Latest revision as of 22:55, 3 December 2016 (view source) Alec (Talk | contribs) m (→‎Statement: Minor change to statement)
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	**** Thus {{M|1=\varphi(g)=\varphi(h)}}		**** Thus {{M|1=\varphi(g)=\varphi(h)}}
	** So whichever representative of {{M|[g]}} we use {{M|\varphi(h)}} for {{M|h\in[g]}} is the same.		** So whichever representative of {{M|[g]}} we use {{M|\varphi(h)}} for {{M|h\in[g]}} is the same.
−	* This is actually all dealt with as a part of [[factor (function)]] not this theorem. However it is worth illustrating.</ref>	+	* This is actually all dealt with as a part of [[factor (function)]] not this theorem. However it is worth illustrating.</ref> and {{M|\bar{\varphi} }} is a [[group homomorphism]].
	Additionally we have {{M|1=\varphi=\overline{\varphi}\circ\pi}} (or in other terms, the diagram on the right commutes)		Additionally we have {{M|1=\varphi=\overline{\varphi}\circ\pi}} (or in other terms, the diagram on the right commutes)

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Latest revision as of 22:55, 3 December 2016

Statement

Let [ilmath]G[/ilmath] and [ilmath]H[/ilmath] be groups, and consider any [ilmath]N\trianglelefteq G[/ilmath] ^{[Note 1]} and [ilmath]\pi:G\rightarrow G/N[/ilmath] the canonical projection of the quotient group, let [ilmath]\varphi:G\rightarrow H[/ilmath] be any group homomorphism, then^[1]:

• 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. If [ilmath]\forall x,y\in X\big[ [w(x)=w(y)]\implies[f(x)=f(y)]\big][/ilmath] then [ilmath]f[/ilmath] "factors (uniquely, if [ilmath]w[/ilmath] is surjective) through [ilmath]w[/ilmath]" via: [ilmath]\bar{f}:W\rightarrow Y[/ilmath] given by [ilmath]\bar{f}:u\mapsto f(w^{-1}(u))[/ilmath] (which is "well defined") Which has the property that [ilmath]f=\bar{f}\circ w[/ilmath] (the diagram on the right commutes)

[ilmath]\xymatrix{X \ar[r]^w \ar[dr]_f & W \ar@{.>}[d]^{\overline{f} } \\ & Y}[/ilmath]

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

1. 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.
2. 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:
     1. [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)
     • 1. [ilmath]\implies\pi(a)\pi(b^{-1})=e[/ilmath]
       2. [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 implies-subset 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.

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

• Group homomorphism theorem (AKA: First group isomorphism theorem)
• Function factorisation theorem

Notes

1. ↑ The notation [ilmath]A\trianglelefteq B[/ilmath] means [ilmath]A[/ilmath] is a normal subgroup of the group [ilmath]B[/ilmath].
2. ↑ 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]
     • So whichever representative of [ilmath][g][/ilmath] we use [ilmath]\varphi(h)[/ilmath] for [ilmath]h\in[g][/ilmath] is the same.
   • This is actually all dealt with as a part of factor (function) not this theorem. However it is worth illustrating.
3. ↑ This is just a nicer way of writing:
   • [ilmath]\bar{\varphi}:g\mapsto \varphi(\pi^{-1}(g))[/ilmath]
4. ↑ 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

1. ↑ Abstract Algebra - Pierre Antoine Grillet

v • d • e   Group theory Overview of the basic and important objects of group theory - a branch of abstract algebra Primitives semigroup [ilmath]\supset[/ilmath] monoid [ilmath]\supset[/ilmath] group [ilmath]\supset[/ilmath] Abelian group (AKA: commutative group), group action Morphisms Group homomorphism, Group isomorphism Subobjects subgroup, normal subgroup, cosets Important theorems First isomorphism theorem, second isomorphism theorem, third isomorphism theorem (overview of the group isomorphism theorems) Important groups Symmetric group (Permutation group), Alternating group, Dihedral group, Quaternion group, Klein-4 group