nyuuzyou/wikis huggingface dataset:
Quotient Theorem for Group Epimorphisms

== Theorem ==
Let $\struct {G, \oplus}$ and $\struct {H, \odot}$ be groups.
Let $\phi: \struct {G, \oplus} \to \struct {H, \odot}$ be a group epimorphism.
Let $e_G$ and $e_H$ be the identities of $G$ and $H$ respectively.
Let $K = \map \ker \phi$ be the kernel of $\phi$.
There exists one and only one group isomorphism $\psi: G / K \to H$ satisfying:
:$\psi \circ q_K = \phi$
where $q_K$ is the quotient epimorphism from $G$ to $G / K$.
== Proof ==
Let $\RR_\phi$ be the equivalence on $G$ defined by $\phi$.
Thus:
:$K = \eqclass {e_G} {\RR_\phi}$
From the Quotient Theorem for Epimorphisms:
:$\RR_\phi$ is compatible with $\oplus$
Thus from Kernel is Normal Subgroup of Domain:
:$K \lhd G$
From Congruence Relation induces Normal Subgroup, $\RR_\phi$ is the equivalence defined by $K$.
Thus, again by Quotient Theorem for Epimorphisms, there is a unique epimorphism $\psi: G / K \to H$ satisfying $\psi \circ q_K = \phi$.
== Also known as ==
Some sources call this the Factor Theorem for Group Epimorphisms.
== Sources ==
* : Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Theorem $12.6$
Category:Quotient Groups
Category:Group Epimorphisms
Category:Quotient Epimorphisms
Category:Group Isomorphisms
Category:Quotient Theorems
Category:Named Theorems
Quotient Theorem for Group Homomorphisms

== Theorem ==
Let $\phi: G \to G'$ be a (group) homomorphism between two groups $G$ and $G'$.
Then $\phi$ can be decomposed into the form:
:$\phi = \alpha \beta \gamma$
where:
:$\alpha: \Img \phi \to G'$ is a monomorphism
:$\beta: G / \map \ker \phi \to \Img \phi$ is an isomorphism
:$\gamma: G \to G / \map \ker \phi$ is an epimorphism.
=== Corollary 1 ===
=== Corollary 2 ===
== Proof ==
=== Monomorphism ===
The mapping $\alpha$ is identified with the inclusion mapping $i: \Img \phi \to G'$ defined as:
:$\forall x \in \Img \phi: \map i x = x$
From Inclusion Mapping is Monomorphism, it follows that $\alpha$ is a monomorphism.
=== Isomorphism ===
From the First Isomorphism Theorem for Groups:
:$\Img \phi \cong G / \map \ker \phi$
for any homomorphism $\phi$.
That is, the image of $\phi$ is isomorphic to the quotient group of $G$ by the kernel of $\phi$.
Thus $\beta$ is such an isomorphism.
=== Epimorphism ===
The mapping $\gamma: G \to G / N$ is given by:
:$\forall a \in G: \map {\gamma} a = a N$
Therefore $\gamma$ is a by definition, a Quotient Group Epimorphism
Furthermore, by Kernel is Normal Subgroup of Domain, the mapping $\gamma: G \to G / \map \ker \phi$ is given by:
:$\forall a \in G: \map \gamma a = a \, \map \ker \phi$
where $a \, \map \ker \phi$ is the left coset of $\map \ker \phi$ by $a$.
From Quotient Epimorphism is Group Epimorphism, it follows that $\gamma$ is such an epimorphism
Hence the result.
This theorem can be illustrated by means of the following commutative diagram:
$\quad\quad \begin{xy}\xymatrix@L+2mu@+1em{
G \ar@{-->}[r]^*{\phi}
\ar[d]^*{\gamma}
&
G'
\\ 
G / \map \ker \phi \ar[r]_*{\beta}^*{\cong}
&
\Img \phi \ar[u]^*{\alpha}
}\end{xy}$
== Also known as ==
The quotient theorem for group homomorphisms is also seen as quotient theorem for groups.
== Examples ==
== Also see ==
* Quotient Theorem for Sets
== Sources ==
* : Chapter $\text{II}$: Groups: Factoring Morphisms: Theorem $4$
* : Chapter $2$: Group Homomorphism and Isomorphism: $\S 67$
Category:Group Homomorphisms
Category:Quotient Groups
Category:Quotient Theorems
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Group Homomorphisms/Corollary 1

== Theorem ==
Let $\struct {G, \odot}$ and $\struct {H, *}$ be groups whose identities are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a group homomorphism.
Let $K$ be the kernel of $\phi$.
Let $N$ be a normal subgroup of $G$.
Let $q_N: G \to \dfrac G N$ denote the quotient epimorphism from $G$ to the quotient group $\dfrac G N$.
Then:
:$N \subseteq K$
:
:there exists a group homomorphism $\psi: \dfrac G N \to H$ such that $\phi = \psi \circ q_N$
This can be illustrated by means of the following commutative diagram:
::$\begin{xy}\xymatrix@L+2mu@+1em{
 G \ar[dr]^*{\phi}
 \ar[d]_*{q}
&
\\ 
 G / N \ar@{-->}[r]_*{\psi}
&
 H 
}\end{xy}$
== Proof 1==
=== Necessary Condition ===
Suppose $\psi$ exists as defined.
Then:
:$K = \phi^{-1} \sqbrk {\set {e_H} } = q^{-1} \sqbrk {\psi^{-1} \sqbrk {\set {e_H} } }$
That is: $\psi^{-1} \sqbrk {\set {e_H} }$ is the kernel of $\psi$.
So by Kernel is Normal Subgroup of Domain, $\psi^{-1} \sqbrk {\set {e_H} }$ is a normal subgroup of $G / N$.
This corresponds via $q^{-1}$ to $K$, which must then be a normal subgroup of $G$ which contains $N$.
=== Sufficient Condition ===
Suppose $N \subseteq K$.
Let $\psi: G / N \to H$ be defined as:
:$\forall g \odot N \in G / N: \map \psi {g \odot N} = \map \phi g \in H$
We have that:
:$n \in g \odot N \implies n^{-1} \odot g \in N \subseteq K$
So:
:$\map \phi {n^{-1} \odot g} = \map \phi n^{-1} * \map \phi g = e_H$
and so:
:$\map \phi n = \map \phi g$.
Thus $\psi$ is well-defined.
Hence $\psi$ is the required group homomorphism.
==Proof 2==
Follows from Universal Property of Quotient Group
== Sources ==
* : Chapter $\text{II}$: Groups: Factoring Morphisms: Corollary
Category:Group Isomorphisms
Category:Quotient Groups
Category:Normal Subgroups
Category:Quotient Theorems
Quotient Theorem for Group Homomorphisms/Corollary 2

== Theorem ==
Let $\struct {G, \odot}$ and $\struct {H, *}$ be groups whose identities are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a group epimorphism.
Let $K$ be the kernel of $\phi$.
Let $N$ be a normal subgroup of $G$.
Let $q_N: G \to G / N$ denote the quotient epimorphism from $G$ to the quotient group $G / N$.
Then:
:$N \subseteq K$
:
:there exists a group epimorphism $\psi: G / N \to H$ such that $\phi = \psi \circ q_N$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.14$
Category:Group Epimorphisms
Category:Quotient Groups
Category:Normal Subgroups
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Group Homomorphisms/Corollary 2/Proof 1

== Theorem ==
== Proof ==
From Quotient Theorem for Group Homomorphisms: Corollary 1:
:$N \subseteq K$
:
:there exists a group homomorphism $\psi: G / N \to H$ such that $\phi = \psi \circ q_N$
From Surjection if Composite is Surjection, it follows that the group homomorphism $\psi$ is a surjection.
Hence by definition, $\psi$ is an epimorphism.
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Group Homomorphisms/Corollary 2/Proof 2

== Theorem ==
== Proof ==
Let $e$ be the identity element of $G$.
Let $\RR$ be the congruence relation defined by $N$ in $G$.
Let $\RR_\phi$ be the equivalence relation induced by $\phi$.
From Condition for Existence of Epimorphism from Quotient Structure to Epimorphic Image:
:there exists an epimorphism $\psi$ from $G / N$ to $H$ which satisfies $\psi \circ q_N = \phi$
:
:$\RR \subseteq \RR_\phi$
It remains to be shown that:
:$\RR \subseteq \RR_\phi$
:
:$N \subseteq K$
== Sources ==
* : Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.18 \ \text {(b)}$
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Group Homomorphisms/Examples

== Examples of Use of Quotient Theorem for Group Homomorphisms ==
=== Real to Complex Numbers under $e^{2 \pi i x}$ ===
=== Inner Automorphism by Inverse Element ===
=== Integers to Modulo Integers under Multiplication ===
=== Integer Power on Circle Group ===
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Group Homomorphisms/Examples/Inner Automorphism by Inverse Element

== Example of Use of Quotient Theorem for Group Homomorphisms ==
Let $G$ be a group.
Let $\Aut G$ denote the automorphism group of $G$.
Let $\phi: G \to \Aut G$ be the homomorphism defined as:
:$\forall g \in G: \map \phi g = \kappa_{g^{-1} }$
where $\kappa_{g^{-1} }$ denotes the inner automorphism of $G$ by $g^{-1}$.
Then $\phi$ can be decomposed into the form:
:$\phi = \alpha \beta \gamma$
in the following way:
:$\alpha: \Inn G \to \Aut G$ is defined as:
::$\forall \kappa \in \Inn G: \map \alpha \kappa = \kappa$
:where $\Inn G$ denotes the inner automorphism group of $G$
:$\beta: G / \map Z G \to \Inn G$ is defined as:
::$\forall g \in G / \map Z G: \map \phi g = \kappa_{g^{-1} }$
:where $G / \map Z G$ denotes the quotient group of $G$ by the center of $G$
:$\gamma: G \to G / \map Z G$ is defined as:
::$\forall g \in G: \map \gamma g = \map {q_{\map Z G} } g = g \, \map Z G$
:where $q_{\map Z G}$ is the quotient epimorphism from $G$ to $G / \map Z G$.
== Proof ==
By definition, $\phi$ is a homomorphism.
From Kernel of Inner Automorphism Group is Center, we have that:
:$\map \ker \phi = \map Z G$
where $\map Z G$ is the center of $G$.
Thus, from the Quotient Theorem for Group Homomorphisms, $\phi$ can be decomposed into:
:$\phi = \alpha \beta \gamma$
where:
:$\alpha: \Inn G \to \Aut G$, which is a monomorphism
:$\beta: G / \map Z G \to \Inn G$, which is an isomorphism
:$\gamma: G \to G / \map Z G$, which is an epimorphism.
The result follows.
== Sources ==
* : Chapter $2$: Group Homomorphism and Isomorphism: $\S 67 \ \beta$
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Group Homomorphisms/Examples/Integer Power on Circle Group

== Example of Use of Quotient Theorem for Group Homomorphisms ==
Let $K$ denote the circle group.
Let $\phi: K \to K$ be the homomorphism defined as:
:$\forall z \in K: \map \phi z = z^n$
for some $n \in \Z_{>0}$.
Then $\phi$ can be decomposed into the form:
:$\phi = \alpha \beta \gamma$
in the following way:
:$\alpha: K \to K$ is defined as:
::$\forall z \in K: \map \alpha z = z$
:that is, $\alpha$ is the identity mapping
:$\beta: S \to K$ is defined as:
::$\forall z \in S: \map \phi z = z^n$
:where $S$ denotes the set defined as:
::$S := \set {z \in \C: z = e^{2 \pi i x}, 0 \le x < \dfrac 1 n}$
:$\gamma: K \to S$ is defined as:
::$\forall z \in K: \map \gamma z = z \bmod \dfrac 1 n$
:where $\bmod$ denotes the modulo operation.
== Proof ==
It is first demonstrated that $\phi$ is a homomorphism:
We have that $1$ is the identity element of $K$, and to confirm:
:$\map \phi 1 = 1^n = 1$
Now we can establish what the kernel of $\phi$ is:
Thus $\map \ker \phi$ is the set of complex $n$th roots of unity:
:$\map \ker \phi = U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$
where $\N_n = \set {0, 1, 2, \ldots, n - 1}$.
Next we establish what the image of $\phi$ is.
Let $w \in \Img \phi$ such that:
:$w = e^{x i}$
for some $x \in \R$.
Thus every element of $K$ has a preimage under $\phi$.
Hence:
:$\Img \phi = K$
Thus, from the Quotient Theorem for Group Homomorphisms, $\phi$ can be decomposed into:
:$\phi = \alpha \beta \gamma$
where:
:$\alpha: K \to K$, which is the identity mapping
:$\beta: K / \map \ker \phi \to K$, which is an isomorphism
:$\gamma: K \to K / \map \ker \phi$, which is an epimorphism.
The result follows.
== Sources ==
* : Chapter $2$: Group Homomorphism and Isomorphism: $\S 67 \ \delta$
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Group Homomorphisms/Examples/Integers to Modulo Integers under Multiplication

== Example of Use of Quotient Theorem for Group Homomorphisms ==
Let $\struct {\Z, +}$ denote the additive group of integers.
Let $\struct {\Z_m, +}$ denote the additive group of integers modulo $m$.
Let $\phi: \struct {\Z, +} \to \struct {\Z_m, +}$ be the homomorphism defined as:
:$\forall k \in \Z: \map \phi k = \eqclass {n k} m$
for some $n \in \Z$.
Let $d := \gcd \set {m, n}$, where $\gcd \set {m, n}$ denotes the GCD of $m$ and $n$.
Let $c := \dfrac m d = \dfrac m {\gcd \set {m, n} }$.
Then $\phi$ can be decomposed into the form:
:$\phi = \alpha \beta \gamma$
in the following way:
:$\alpha: \struct {d \, \Z_c, +} \to \struct {\Z_m, +}$ is defined as:
::$\forall x \in d \, \Z_c: \map \alpha x = x$
:where $d \, \Z_c := \set {0, d, 2 d, \ldots, \paren {c - 1} d}$
:$\beta: \Z_c \to d \, \Z_c$ is defined as:
::$\forall \eqclass x c \in \Z_c: \map \beta {\eqclass x c} = \eqclass {n x} m$
:$\gamma: \Z \to \Z_c$ is defined as:
::$\forall x \in \Z: \map \gamma x = \eqclass {x \bmod c} c$
:where $\bmod$ denotes the modulo operation.
== Proof ==
It is first demonstrated that $\phi$ is a homomorphism:
We have that:
By Group Homomorphism Preserves Identity it is confirmed that $\eqclass 0 m$ is the identity of $\struct {\Z_m, \times}$.
Now we can establish what the kernel of $\phi$ is:
That is:
Next we establish what the image of $\phi$ is:
Thus, from the Quotient Theorem for Group Homomorphisms, $\phi$ can be decomposed into:
:$\phi = \alpha \beta \gamma$
where:
:$\alpha: d \, \Z_c \to \Z_m$, which is a monomorphism
:$\beta: \Z / c \, \Z \to d \, \Z_c$, which is an isomorphism
:$\gamma: \Z \to \Z / c \, \Z$, which is an epimorphism.
Finally we have from Quotient Group of Integers by Multiples:
:$\Z / c \, \Z = \Z_c $
and the result follows.
== Sources ==
* : Chapter $2$: Group Homomorphism and Isomorphism: $\S 67 \ \gamma$
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Group Homomorphisms/Examples/Real to Complex Numbers under e^2 pi i x

== Example of Use of Quotient Theorem for Group Homomorphisms ==
Let $\struct {\R, +}$ denote the additive group of real numbers.
Let $\struct {\C_{\ne 0}, \times}$ denote the multiplicative group of complex numbers.
Let $\phi: \struct {\R, +} \to \struct {\C_{\ne 0}, \times}$ be the homomorphism defined as:
:$\forall x \in \R: \map \phi x = e^{2 \pi i x}$
Then $\phi$ can be decomposed into the form:
:$\phi = \alpha \beta \gamma$
in the following way:
:$\alpha: \struct {K, \times} \to \struct {\C_{\ne 0}, \times}$ is defined as:
::$\forall z \in K: \map \alpha z = z$
:where $\struct {K, \times}$ denotes the circle group:
::$K = \set {z \in \C: \cmod z = 1}$
::$\times$ is the operation of complex multiplication
:$\beta: \hointr 0 1 \to K$ is defined as:
::$\forall x \in \hointr 0 1: \map \beta x = e^{2 \pi i x}$
:where $\hointr 0 1$ denotes the right half-open real interval $\set {x \in \R: 0 \le x < 1}$
:$\gamma: \R \to \hointr 0 1$ is defined as:
::$\forall x \in \R: \map \gamma x = \fractpart x$
:where $\fractpart x$ is the fractional part of $x$:
::$\fractpart x := x - \floor x$
== Proof ==
It is first demonstrated that $\phi$ is a homomorphism:
We have that:
By Group Homomorphism Preserves Identity it is confirmed that $1$ is the identity of $\struct {\C_{\ne 0}, \times}$.
Now we can establish what the kernel of $\phi$ is:
Thus:
:$\map \ker \phi = \Z$
where $\Z$ denotes the set of integers.
Next we establish what the image of $\phi$ is:
Thus, from the Quotient Theorem for Group Homomorphisms, $\phi$ can be decomposed into:
:$\phi = \alpha \beta \gamma$
where:
:$\alpha: K \to \C_{\ne 0}$, which is a monomorphism
:$\beta: \R / \Z \to K$, which is an isomorphism
:$\gamma: \R \to \R / \Z$, which is an epimorphism.
As $\beta$ is an isomorphism, $\beta$ is also a bijection and so $\R / \Z = \Preimg \beta$ can be deduced:
We have specifically selected $\hointr 0 {2 \pi}$ as the image of the principal argument of $\Ln z$.
Other half-open real intervals whose length is $2 \pi$ work equally well, for example $\hointl {-\pi} \pi$.
Thus:
Thus we have:
:$\Preimg \beta = \R / \Z = \hointr 0 1$
and the result follows.
== Sources ==
* : Chapter $\text I$: Sets and Functions: Exercise $\text L$
* : Chapter $2$: Group Homomorphism and Isomorphism: $\S 67 \ \alpha$
Category:Quotient Theorem for Group Homomorphisms
Quotient Theorem for Monomorphisms

== Theorem ==
Let $K, L$ be fields of quotients of integral domains $\struct {R, +_R, \circ_R}, \struct {S, +_S, \circ_S}$ respectively.
Let $\phi: R \to S$ be a monomorphism.
Then there is one and only one monomorphism $\psi: K \to L$ extending $\phi$, and:
:$\forall x \in R, y \in R^*: \map \psi {\dfrac x y} = \dfrac {\map \phi x} {\map \phi y}$
Also, if $\phi$ is a ring isomorphism, then so is $\psi$.
== Proof ==
By definition, $\struct {K, \circ_R}$ and $\struct {L, \circ_S}$ are inverse completions of $\struct {R, \circ_R}$ and $\struct {S, \circ_S}$ respectively.
So by the Extension Theorem for Homomorphisms, there is one and only one monomorphism $\psi: \struct {K, \circ_R} \to \struct {L, \circ_S$ extending $\phi$.
Thus:
:$\forall x \in R, y \in R^*: \map \psi {\dfrac x y} = \dfrac {\map \phi x} {\map \phi y}$
By the Extension Theorem for Isomorphisms, $\psi$ is an isomorphism if $\phi$ is.
Thus, $\forall x, y \in R, z, w \in R^*$:
Thus $\psi: K \to L$ is a monomorphism.
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.10$
Category:Fields of Quotients
Category:Integral Domains
Category:Ring Monomorphisms
Category:Named Theorems
Quotient Theorem for Sets

== Theorem ==
A mapping $f: S \to T$ can be uniquely factored into a surjection, followed by a bijection, followed by an injection.
Thus:
:$f = i \circ r \circ q_{\RR_f}$
where:
where:
:$\RR_f$ is the equivalence induced by $f$
:$S / \RR_f$ is the quotient set of $S$ induced by $\RR_f$
This can be illustrated using a commutative diagram as follows:
$\quad\quad \begin {xy} \xymatrix@L + 2mu@ + 1em {
S \ar@{-->}[rrr]^*{f = i_T \circ r \circ q_{\RR_f} }
\ar[d]_*{q_{\RR_f} }
& & & T \\
S / \RR_f \ar[rrr]_*{r} & & & \Img f \ar[u]_*{i_T} } \end {xy}$
== Proof ==
== Also known as ==
Otherwise known as the factoring theorem or factor theorem.
This construction is known as the canonical decomposition of $f$.
== Examples ==
== Also see ==
* Factoring Mapping into Quotient and Injection
* Factoring Mapping into Surjection and Inclusion
* Quotient Theorem for Surjections
== Sources ==
* : Chapter $\text{I}$: Sets and Functions: Factoring Functions: Theorem $11$
Category:Mapping Theory
Category:Quotient Mappings
Category:Quotient Sets
Category:Named Theorems
Category:Quotient Theorems
Category:Quotient Theorem for Sets
Quotient Theorem for Sets/Examples/Real Square Function

== Example of Use of Quotient Theorem for Sets ==
Let $f: \R \to \R$ denote the square function:
:$\forall x \in \R: \map f x = x^2$
We define $\RR_f \subseteq S \times S$ to be the relation:
:$\tuple {x_1, x_2} \in \RR_f \iff {x_1}^2 = {x_2}^2$
that is:
:$x_1 \mathrel {\RR_f} x_2 \iff x_1 = \pm x_2$
The quotient set of $\R$ induced by $\RR_f$ is thus the set $\R / \RR_f$ of $\RR$-classes of $\RR$:
Hence the quotient mapping $q_{\RR_f}$:
:$q_{\RR_f}: \R \to \R / \RR_f: \map {q_{\RR_f} } x = \eqclass x {\RR_f}$
puts $x$ into its equivalence class $\set {x, -x}$.
We note in passing that $\eqclass x {\RR_f}$ has $2$ elements unless $x = 0$.
The renaming mapping is defined as:
:$r: \R / \RR_f \to \Img f: \map r {\eqclass x {\RR_f} } = x^2$
where $\Img f = \R_{\ge 0}$.
Finally the inclusion mapping is defined as:
:$i_{\R_{\ge 0} }: \R_{\ge 0} \to \R: \map {i_{\R_{\ge 0} } } x = x$
== Sources ==
* : Chapter $\text{I}$: Sets and Functions: Factoring Functions
Category:Quotient Theorem for Sets
Quotient Theorem for Sets/Proof

== Theorem ==
== Proof ==
From Factoring Mapping into Surjection and Inclusion, $f$ can be factored uniquely into:
:A surjection $g: S \to \Img f$, followed by:
:The inclusion mapping $i: \Img f \to T$ (an injection).
$\quad\quad \begin{xy}\xymatrix@L+2mu@+1em {
S \ar[drdr]_*{g}
\ar@{-->}[rr]^*{f = i_T \circ g}
& & T
\\ \\
& & \Img f \ar[uu]_*{i_T}
}\end{xy}$
From the Quotient Theorem for Surjections, the surjection $g$ can be factored uniquely into:
:The quotient mapping $q_{\RR_f}: S \to S / \RR_f$ (a surjection), followed by:
:The renaming mapping $r: S / \RR_f \to \Img f$ (a bijection).
Thus:
:$f = i_T \circ \paren {r \circ q_{\RR_f} }$
As Composition of Mappings is Associative it can be seen that $f = i_T \circ r \circ q_{\RR_f}$.
The commutative diagram is as follows:
$\quad\quad \begin {xy} \xymatrix@L + 2mu@ + 1em {
S \ar@{-->}[rrr]^*{f = i_T \circ r \circ q_{\RR_f} }
\ar[ddd]_*{q_{\RR_f} }
\ar@{..>}[drdrdr]_*{g = r \circ q_{\RR_f} }
& & & T \\ \\ \\
S / \RR_f \ar[rrr]_*{r} & & & \Img f \ar[uuu]_*{i_T} } \end {xy}$
== Also known as ==
Otherwise known as the factoring theorem or factor theorem.
This construction is known as the canonical decomposition of $f$.
== Also see ==
* Factoring Mapping into Quotient and Injection
* Factoring Mapping into Surjection and Inclusion
* Quotient Theorem for Surjections
== Sources ==
* : Chapter $\text{I}$: Sets and Functions: Exercise $\text{K}$
* : $\S 6$. Indexed families; partitions; equivalence relations: Theorem $6.6$
Category:Quotient Theorem for Sets
Quotient Theorem for Surjections

== Theorem ==
Let $f: S \to T$ be a surjection.
Then there is one and only one bijection $r: S / \RR_f \to T$ such that:
:$r \circ q_{\RR_f} = f$
where:
:$\RR_f$ is the equivalence induced by $f$
:$r: S / \RR_f \to T$ is the renaming mapping
:$q_{\RR_f}: S \to S / \RR_f$ is the quotient mapping induced by $\RR_f$.
This can be illustrated using a commutative diagram as follows:
$\quad\quad \begin {xy} \xymatrix@L + 2mu@ + 1em {
S \ar@{-->}[rr]^*{f = r \circ q_{\RR_f} }
\ar[dd]_*{q_{\RR_f} }
&& T \\
\\
S / \RR_f \ar[urur]_*{r} 
} \end {xy}$
== Proof ==
From the definition of Induced Equivalence, the mapping $f: S \to T$ induces an equivalence $\RR_f$ on its domain.
As $f: S \to T$ is a surjection, $T = \Img f$ by definition.
From Renaming Mapping is Bijection, the renaming mapping $r: S / \RR_f \to T$ is a bijection, where $S / \RR_f$ is the quotient set of $S$ by $\RR_f$.
Hence:
:$r \circ q_{\RR_f} = f$
:$r$ is the only mapping $r: S / \RR_f \to T$ that satisfies this equality.
== Also known as ==
Also known as the factor theorem for surjections.
== Also see ==
* Factoring Mapping into Quotient and Injection
* Factoring Mapping into Surjection and Inclusion
* Quotient Theorem for Sets
== Sources ==
* : Chapter $\text {II}$: New Structures from Old: $\S 10$: Equivalence Relations: Theorem $10.5$
* : $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.4 \ \text{(b)}$
Category:Surjections
Category:Quotient Sets
Category:Quotient Mappings
Category:Quotient Theorems
Category:Named Theorems
Quotient Topological Vector Space is Hausdorff iff Linear Subspace is Closed

== Theorem ==
Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $N$ be a linear subspace of $X$.
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Let $\tau_N$ be the quotient topology on $X/N$.
Then $\struct {X/N, \tau_N}$ is Hausdorff :
:$N$ is a closed linear subspace.
== Proof 1 ==
== Proof 2 ==
 
Category:Quotient Topological Vector Spaces
Category:Hausdorff Topological Vector Spaces
Category:Quotient Topological Vector Space is Hausdorff iff Linear Subspace is Closed
Quotient Topological Vector Space is Hausdorff iff Linear Subspace is Closed/Proof 1

== Theorem ==
== Proof ==
Let $\pi : X \to X/N$ be the quotient mapping.
=== Necessary Condition ===
Suppose that $\struct {X/N, \tau_N}$ is Hausdorff.
From Characterization of Hausdorff Topological Vector Space, $\set { {\mathbf 0}_{X/N} }$ is closed in $\struct {X/N, \tau_N}$.
From the definition of the quotient topology, $\pi$ is continuous.
Hence $\pi^{-1} \sqbrk {\set { {\mathbf 0}_{X/N} } }$ is closed in $\struct {X, \tau}$.
From Kernel of Quotient Mapping, we have $\pi^{-1} \sqbrk {\set { {\mathbf 0}_{X/N} } } = N$.
Hence $N$ is closed in $\struct {X, \tau}$.
=== Sufficient Condition ===
Suppose that $N$ is closed in $\struct {X, \tau}$.
Then $X \setminus N$ is open in $\struct {X, \tau}$.
From Quotient Mapping on Quotient Topological Vector Space is Open Mapping, $\pi$ is an open mapping.
Hence $\pi \sqbrk {X \setminus N}$ is open in $\struct {X/N, \tau_N}$.
Since $\pi$ is surjective by construction and $\pi^{-1} \sqbrk {\set { {\mathbf 0}_{X/N} } } = N$, we have:
:$\pi \sqbrk {X \setminus N} = \paren {X/N} \setminus \set { {\mathbf 0}_{X/N} }$
So $\paren {X/N} \setminus \set { {\mathbf 0}_{X/N} }$ is open in $\struct {X/N, \tau_N}$.
So $\set { {\mathbf 0}_{X/N} }$ is closed in $\struct {X/N, \tau_N}$.
From Characterization of Hausdorff Topological Vector Space, we have that $\struct {X/N, \tau_N}$ is Hausdorff.
 
Category:Quotient Topological Vector Space is Hausdorff iff Linear Subspace is Closed