nyuuzyou/wikis huggingface dataset:
Keith Number/Examples

== Examples of Keith Numbers == 
The following are examples of Keith numbers:
=== Keith Number: $14$ ===
=== Keith Number: $19$ ===
=== Keith Number: $197$ ===
=== Keith Number: $129 \, 572 \, 008$ ===
=== Keith Number: $251 \, 133 \, 297$ ===
=== Keith Number: $754 \, 788 \, 753 \, 590 \, 897$ ===
Category:Keith Numbers
Keith Number/Examples/129,572,008

== Examples of Keith Number ==
$129 \, 572 \, 008$ is a Keith number:
:$1, 2, 9, 5, 7, 2, 0, 0, 8, \ldots, 32 \, 456 \, 930, 64 \, 849 \, 899, 129 \, 572 \, 008, \ldots$
== Proof ==
By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\left({1, 2, 9, 5, 7, 2, 0, 0, 8}\right)$:
Thus $129 \, 572 \, 008$ occurs in $K$ and the result follows by definition of Keith number.
== Historical Note ==
== Sources ==
* 
* : $129,572,008$
Category:Keith Numbers
Category:129,572,008
Keith Number/Examples/129,572,008/Historical Note

== Historical Note on Keith Number/Examples/129,572,008|Keith Number $129 \, 572 \, 008$ ==
The Keith number $129 \, 572 \, 008$ was discovered, along with $251 \, 233 \, 297$, by , who reported it in his work of $1991$.
== Sources ==
* : $129,572,008$
Category:Historical Notes
Keith Number/Examples/14

== Examples of Keith Number ==
$14$ is a Keith number:
:$1, 4, 5, 9, 14, \ldots$
== Proof ==
By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\left({1, 4}\right)$:
Thus $14$ occurs in $K$ and the result follows by definition of Keith number.
== Sources ==
* : $14$
Category:Keith Numbers
Keith Number/Examples/19

== Examples of Keith Number ==
$19$ is a Keith number:
:$1, 9, 10, 19, \ldots$
== Proof ==
By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\left({1, 9}\right)$:
Thus $19$ occurs in $K$ and the result follows by definition of Keith number.
Category:Keith Numbers
Keith Number/Examples/197

== Examples of Keith Number ==
$197$ is a Keith number:
:$1, 9, 7, 17, 33, 57, 107, 197, \ldots$
== Proof ==
By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\tuple {1, 9, 7}$:
Thus $197$ occurs in $K$ and the result follows by definition of Keith number.
Category:Keith Numbers
Keith Number/Examples/251,133,297

== Examples of Keith Number ==
$251 \, 133 \, 297$ is a Keith number:
:$2, 5, 1, 1, 3, 3, 2, 9, 7, 33, 64, 123, 245, 489, 975, 1947, 3892, 7775,$
:$15 \, 543, 31 \, 053, 62 \, 042, 123 \, 961, 247 \, 677, 494 \, 865, 988 \, 755,$
:$1 \, 975 \, 563, 3 \, 947 \, 234, 7 \, 886 \, 693, 15 \, 757 \, 843, 31 \, 484 \, 633,$
:$62 \, 907 \, 224, 125 \, 690 \, 487, 251 \, 133 \, 297, \ldots$
== Proof ==
By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\left({2, 5, 1, 1, 3, 3, 2, 9, 7}\right)$:
Thus $251 \, 233 \, 297$ occurs in $K$ and the result follows by definition of Keith number.
== Historical Note ==
== Sources ==
* 
* : $129,572,008$
Category:Keith Numbers
Category:251,133,297
Keith Number/Examples/754,788,753,590,897

== Examples of Keith Number ==
$754 \, 788 \, 753 \, 590 \, 897$ is a Keith number:
:$7, 5, 4, 7, 8, 8, 7, 5, 3, 5, 9, 0, 8, 9, 7, \ldots, $
== Proof ==
By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\paren {7, 5, 4, 7, 8, 8, 7, 5, 3, 5, 9, 0, 8, 9, 7}$:
Thus $754 \, 788 \, 753 \, 590 \, 897$ occurs in $K$ and the result follows by definition of Keith number.
== Sources ==
* 
* : $754,788,753,590,897$
Category:Keith Numbers
Category:754,788,753,590,897
Kelley's Theorem

== Theorem ==
Let $\struct {D, \preceq}$ be a directed set,
Let $S$ be a non-empty set.
Let $n: D \to S$ be a net in $S$.
Then $n$ has a universal subnet.
== Proof ==
== Also see ==
* Kelley's Theorem is Equivalent to Axiom of Choice
Category:Set Theory
Category:Nets (Set Theory)
Category:Equivalents of Axiom of Choice
Kelvin-Stokes Theorem

== Theorem ==
Let $S$ be some orientable smooth surface with boundary in $\R^3$.
Let $\mathbf F:\R^3 \to \R^3$ be a vector-valued function with Euclidean coordinate expression:
:$\mathbf F = f_1 \mathbf i + f_2 \mathbf j + f_3 \mathbf k$
where $f_i: \R^3 \to \R$.
Then:
:$\ds \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z = \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A$
where $\mathbf n$ is the unit normal to $S$ and $\d A$ is the area element on the surface.
== Proof ==
Let $\mathbf r:\R^2 \to \R^3, \map {\mathbf r} {s, t}$ be a smooth parametrization of $S$ from some region $R$ in the $st$-plane, so that:
:$\map {\mathbf r} R = S$
and:
:$\map {\mathbf r} {\partial R} = \partial S$
First, we convert the into a line integral:
so that if we define:
:$\mathbf G = \paren {G_1, G_2} = \paren {\mathbf F \cdot \dfrac {\partial \mathbf r} {\partial s}, \mathbf F \cdot \dfrac {\partial \mathbf r} {\partial t} }$
then:
:$\ds \int_{\partial S} \mathbf F \cdot \rd \mathbf r = \int_{\partial R} \mathbf G \cdot \rd \mathbf s$
where $\mathbf s$ is the position vector in the $s t$-plane. 
Note that:
We turn now to the expression on the and write it in terms of $s$ and $t$:
Let us investigate the integrand:
 
 
That is:
:$\ds \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A = \iint_R \paren {\frac {\partial G_2} {\partial s} - \frac {\partial G_1} {\partial t} } \rd s \rd t$
By Green's Theorem, this can be written as:
:$\ds \int_{\partial R} \mathbf G \cdot \rd \mathbf s$
Hence both sides of the theorem equation are equal.
== Also known as ==
== Historical Note ==
== Sources ==
* : $\S 22$: Integrals involving Vectors: $22.60$
* 
* 
* 
Category:Kelvin-Stokes Theorem
Category:Stokes' Theorem
Category:Integral Calculus
Kelvin-Stokes Theorem/Also known as

== Kelvin-Stokes Theorem: Also known as == 
The Kelvin-Stokes theorem is also known as the classical Stokes' theorem.
It is also known as just Stokes's theorem, or Stokes' theorem.
== Sources ==
* : $\S 22$: Integrals involving Vectors: $22.60$
* 
* 
* 
Category:Kelvin-Stokes Theorem
Kendall's Coefficient of Concordance/Examples/Arbitrary Example 1

== Example of Kendall's Coefficient of Concordance ==
Consider the $3$ competitors $\text {Xavier}$, $\text {Yuri}$ and $\text {Zal}$, who are demonstrating their skills to the $4$ judges $\text {Araminta}$, $\text {Boecluvius}$, $\text {Coriolanius}$ and $\text {Derek}$.
The following table shows the ranking afforded the competitors by each judge, with the row summations per competitor:
Kendall's coefficient of concordance is $\dfrac 1 {16}$.
== Proof ==
The mean $M$ is given by:
:$M = \dfrac 1 2 \times 4 \times \paren {3 + 1} = 8$
The $S$ value is given by:
Hence Kendall's coefficient of concordance is shown to be:
== Sources ==
* 
* 
Category:Examples of Kendall's Coefficient of Concordance
Kendall's Coefficient of Concordance/Examples/Arbitrary Example 2

== Example of Kendall's Coefficient of Concordance ==
Consider the $4$ competitors $\text {Wilhelmina}$, $\text {Xanthippe}$, $\text {Yondalla}$ and $\text {Zena}$, who are demonstrating their skills to the $4$ judges $\text {Ariadne}$, $\text {Boudicca}$, $\text {Constantine}$ and $\text {Donald}$.
The following table shows the ranking afforded the competitors by each judge, with the row summations per competitor:
Kendall's coefficient of concordance is $0$.
== Proof ==
The mean $M$ is given by:
:$M = \dfrac 1 2 \times 4 \times \paren {4 + 1} = 10$
The $S$ value is given by:
Hence Kendall's coefficient of concordance is shown to be:
== Sources ==
* 
* 
Category:Examples of Kendall's Coefficient of Concordance
Kendall's Coefficient of Concordance is between 0 and 1

== Theorem ==
Kendall's coefficient of concordance $W$ fulfils the following inequality:
:$0 \le W \le 1$
where:
:$0$ indicates complete disagreement
:$1$ indicates complete agreement.
== Proof ==
== Sources ==
* 
* 
Category:Kendall's Coefficient of Concordance
Kepler's Conjecture

== Theorem ==
The densest packing of identical spheres in space is obtained when the spheres are arranged with their centers at the points of a face-centered cubic lattice.
This obtains a density of $\dfrac \pi {3 \sqrt 2} = \dfrac \pi {\sqrt {18} }$:
:$\dfrac \pi {\sqrt {18} } = 0 \cdotp 74048 \ldots$
== Proof ==
== Also known as ==
== Historical Note ==
== Sources ==
* 
* : $0 \cdotp 7404 \ldots$
* : $0 \cdotp 7404 \ldots$
* 
* 
Category:Kepler's Conjecture
Category:Hilbert 23
Category:Spheres
Kepler's Conjecture/Historical Note

== Historical Note on Kepler's Conjecture ==
This result was conjectured by in $1611$.
While it is in a certain sense obvious that the most efficient technique for packing spheres is the one traditionally used by greengrocer's to stack orange's, it proved challenging to actually prove it.
:Many mathematicians believe, and all physicists know, that the density cannot exceed $\dfrac \pi {\sqrt {18} }$.
:::: -- 
The proof by was finally declared complete and correct in $2014$, at the climax of a project spanning some $20$ years.
== Sources ==
* 
* : $0 \cdotp 7404 \ldots$
* : $0 \cdotp 7404 \ldots$
* 
* 
Category:Historical Notes
Kepler's Conjecture/Historical Note/Mistake

== Source Work ==
:
:Kepler's conjecture (J. Kepler, 1611)
== Mistake ==
:The conjecture was proved in $2006$ by .
== Correction ==
Technically incorrect.
 accepted the proof presented by in $2005$, but there existed only $99\%$ certainty that it was correct.
The proof was finally declared complete and correct in August $2014$.
== Sources ==
* 
Category:Mistakes/The Penguin Dictionary of Mathematics
Kepler's Conjecture/Mistake

== Source Work ==
:
:Thème et variations
::$0,77963 55700 \ldots$
== Mistake ==
* $\sqrt {18} \paren {\operatorname {Arcos} 1/3 - \pi / 3}$
:Le meilleur majorant connu pour la densité d'un empilement de sphères dans $R^3$.
That is, in English:
* $\sqrt {18} \paren {\arccos 1/3 - \pi / 3}$
:The best known upper bound for the density of a stack of spheres in $\R^3$.
== Correction ==
This constant is in fact the packing density of a regular tetrahedron.
That is:
:Let $S$ be a regular tetrahedron of edge length $2$.
:Let $B$ be the part of $S$ that lies within distance $1$ of some vertex.
:Then this constant is the ratio of the volume of $B$ to the volume of $S$.
== Also see ==
* Packing Density of Regular Tetrahedron
== Sources ==
* : $0,77963 55700 \ldots$
Category:Mistakes/Les Nombres Remarquables
Kepler's Explanation for Spacing of Planets

== Conjecture ==
In the 16th century there were $6$ known planets.
It was also known that there were Five Platonic Solids.
 reasoned that it was possible to fit the orbits of the planets around the sun into spheres such that:
:$(1): \quad$ Between each sphere was a (real or imaginary) platonic solid
:$(2): \quad$ The sphere of one planet was inscribed within the platonic solid between that and the next planet out
:$(3): \quad$ The sphere of one planet was circumscribed around the platonic solid between that and the next planet in.
Thus:
:Between the orbits of Mercury and Venus was an octahedron
:Between the orbits of Venus and Earth was an icosahedron
:Between the orbits of Earth and Mars was a dodecahedron
:Between the orbits of Mars and Jupiter was a tetrahedron
:Between the orbits of Jupiter and Saturn was a cube.
== Historical Note ==
== Sources ==
* : $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
* : $5$
* 
* : Chapter $\text {A}.10$: Kepler ()
* : $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
* : $5$
* : Chapter $8$: The System of the World: Kepler
Category:Famous False Conjectures
Category:Celestial Mechanics
Category:Solar System
Kepler's Explanation for Spacing of Planets/Historical Note

== Historical Note on Kepler's Explanation for Spacing of Planets ==
's conjecture on the spacing of the planets was made in his $1596$ work .
:The earth's orbit is the measure of all things; circumscribe around it a dodecahedron, and the circle containing this will be Mars: circumscribe around Mars a tetrahedron, and the circle containing this will be Jupiter: circumscribe around Jupiter a cube, and the circle containing this will be Saturn. Now inscribe within the earth an icosahedron, and the circle contained in it will be Venus; inscribe within Venus an octahedron, and the circle contained within it will be Mercury. You now have the reason for the number of planets.
His model, however ingenious, did not agree well with observations, particularly in light of subsequent, more accurate, observational data.
However, it made his reputation as an imaginative thinker and keen astronomer, and importantly this was one of the first theses based on a heliocentric universe.
He brought him to the attention of the scientific community, including , and soon he was offered a position working for .
== Sources ==
* : $5$
* : Chapter $\text {A}.10$: Kepler ()
* : $5$
* : Chapter $8$: The System of the World: Kepler
Category:Historical Notes
Kepler's Laws of Planetary Motion

== Physical Laws ==
Kepler's laws of planetary motion are a set of three physical laws that describe the motion of planets in the solar system.
=== First Law ===
=== Second Law ===
=== Third Law ===
== Also known as ==
== Historical Note ==
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation
* 
* 
* : Chapter $8$: The System of the World: Kepler
* 
Category:Kepler's Laws of Planetary Motion
Category:Celestial Mechanics
Kepler's Laws of Planetary Motion/Also known as

== Kepler's Laws of Planetary Motion: Also known as == 
Kepler's Laws of Planetary Motion are also known as just:
:Kepler's Laws
:the Laws of Planetary Motion.
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* 
* 
* 
* 
Category:Kepler's Laws of Planetary Motion
Kepler's Laws of Planetary Motion/Historical Note

== Historical Note ==
 derived his three laws of planetary motion in the early $1600$s from a concentrated study over the course of $20$ years of the colossal wealth of observational data which had been made previously by of the behavior of the planets of the solar system, and in particular Mars.
The first two of these results he published in his gigantic $1609$ work .
The third appears some ten years later in his of $1619$.
It was who managed to interpret these three laws and so work out what is now known as Newton's Law of Universal Gravitation, from which Kepler's laws can straightforwardly be derived.
== Sources ==
* : $\S 3.21$: Newton's Law of Gravitation: Footnote
* : Chapter $\text {A}.10$: Kepler ()
* 
* 
* : Chapter $8$: The System of the World: Kepler
Category:Historical Notes
Kepler's Laws of Planetary Motion/Second Law

== Physical Law ==
Kepler's second law of planetary motion is one of the three physical laws of celestial mechanics deduced by :
:The line joining the sun and a planet sweeps out equal areas in equal times.
:
== Proof ==
Consider a planet $p$ of mass $m$ moving around the sun in the plane under the influence of the force $\mathbf F$ imparted by the gravitational field which the two bodies give rise to.
Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$.
By definition of the gravitational field, $\mathbf F$ is a central force.
From Derivative of Angular Component under Central Force:
:$r^2 \dfrac {\d \theta} {\d t} = h$
for some constant $h$.
, assume that $h > 0$, which means that $p$ is travelling in the direction of positive $\theta$.
Let $\map A t$ be the area swept out by $\mathbf r$ in time $t$ relative to some fixed point of reference.
For a small angle $\delta \theta$, the area $\delta A$ can be approximated to the area of a sector of a circle.
Thus:
:$\delta A = \dfrac {r^2 \delta \theta} 2$
and so in the limit:
That is, given a time interval $t_2 - t_1$, the area $\map A {t_2} - \map A {t_1}$ is the same, whatever the physical position of $p$.
== Also see ==
* Kepler's First Law of Planetary Motion
* Kepler's Third Law of Planetary Motion
== Historical Note ==
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : $\S 3.21$: Newton's Law of Gravitation: $(10)$
* : Chapter $\text {A}.10$: Kepler ()
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation
* 
* 
* : Chapter $8$: The System of the World: Kepler
* 
Category:Kepler's Laws of Planetary Motion
Kepler's Laws of Planetary Motion/Third Law

== Physical Law ==
Kepler's third law of planetary motion is one of the three physical laws of celestial mechanics deduced by :
:The square of the period of the orbit of a planet around the sun is proportional to the cube of its mean distance from the sun.
== Proof ==
Consider a planet $p$ of mass $m$ orbiting a star $S$ of mass $M$ under the influence of the gravitational field which the two bodies give rise to.
From Kepler's First Law of Planetary Motion, $p$ travels in an elliptical orbit around $S$:
:$(1): \quad r = \dfrac {h^2 / k} {1 + e \cos \theta}$
where $k = G M$.
From Equation of Ellipse in Reduced Form: Cartesian Frame, the equation of the orbit can also be given as:
:$(2): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
where the foci are placed at $\tuple {\pm c, 0}$.
From Focus of Ellipse from Major and Minor Axis:
:$a^2 - b^2 = c^2$
and also:
:$e = \dfrac c a$
Thus:
:
The mean distance $a$ of $p$ from the focus $F$ is half the sum of the least and greatest values of $r$.
So $(1)$ and $(3)$ give:
Let $T$ be the orbital period of $p$.
From Area of Ellipse, the area $\AA$ of the orbit is given by:
:$\AA = \pi a b$
From Kepler's Second Law of Planetary Motion it follows that:
:$\dfrac {h T} 2 = \pi a b$
and so from $(4)$:
We have that:
:$k = G M$
where $G$ is the universal gravitational constant and $M$ is the mass of $S$.
Hence the result.
== Astronomical Units ==
== Examples ==
== Also see ==
* Kepler's First Law of Planetary Motion
* Kepler's Second Law of Planetary Motion
== Historical Note ==
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : $\S 3.21$: Newton's Law of Gravitation: $(22)$
* : Chapter $\text {A}.10$: Kepler ()
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation
* 
* 
* : Chapter $8$: The System of the World: Kepler
* 
Category:Kepler's Laws of Planetary Motion
Kepler's Laws of Planetary Motion/Third Law/Astronomical Units

== Physical Law ==
When distance is measured in astronomical units and time is measured in years, Kepler's Third Law of Planetary Motion can be expressed as:
:$T^2 = a^3$
These units are derived specifically from the nature of the revolution of the Earth in its orbit, for which $T = 1 \, \mathrm{yr}$ and $a = 1 \, \mathrm {au}$.
== Sources ==
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation
Category:Kepler's Laws of Planetary Motion
Kepler's Laws of Planetary Motion/Third Law/Examples

== Examples of Use of Kepler's Third Law of Planetary Motion ==
Let $P$ be a planet orbiting the sun $S$.
Let $P$ be:
:$\text{(a)}: \quad$ Twice as far away from $S$ as the Earth
:$\text{(b)}: \quad$ $3$ times as far away from $S$ as the Earth
:$\text{(c)}: \quad$ $25$ times as far away from $S$ as the Earth.
Then the orbital period of $P$ is:
:$\text{(a)}: \quad$ approximately $2.8$ years
:$\text{(b)}: \quad$ approximately $5.2$ years
:$\text{(c)}: \quad$ $125$ years.
== Proof ==
Let the orbital period of Earth be $T'$ years.
Let the mean distance of Earth from $S$ be $A$.
Let the orbital period of $P$ be $T$ years.
Let the mean distance of $P$ from $S$ be $a$.
By Kepler's Third Law of Planetary Motion:
Thus the required orbital periods are:
:$\text{(a)}: \quad 2^{3/2} = 2 \sqrt 2 \approx 2.8$ years
:$\text{(b)}: \quad 3^{3/2} = 3 \sqrt 3 \approx 5.2$ years
:$\text{(c)}: \quad 25^{3/2} = 125$ years.
== Sources ==
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation: Problem $2$
Category:Kepler's Laws of Planetary Motion
Kernel Transformation of Measure is Measure

== Theorem ==
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $N: X \times \Sigma \to \overline \R_{\ge0}$ be a kernel.
Then $\mu N: X \to \overline \R$, the kernel transformation of $\mu$, is a measure.
== Proof ==
From the definition of the kernel transformation of $\mu$, we have: 
:$\ds \map {\paren {\mu N} } E = \int \map N {x, E} \rd \map \mu x$
for each $E \in \Sigma$. 
We verify each of the conditions for a measure in turn. 
=== Proof of $(1)$ ===
Let $E \in \Sigma$. 
From the definition of a kernel, we have that: 
:$x \mapsto \map N {x, E}$ is a positive $\Sigma$-measurable function. 
From the definition of the $\mu$-integral of a positive $\Sigma$-measurable function, we have: 
:$\ds \int \map N {x, E} \rd \map \mu x \ge 0$
So:
:$\map {\paren {\mu N} } E \ge 0$
for each $E \in \Sigma$, verifying $(1)$. 
=== Proof of $(2)$ ===
Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of pairwise disjoint $\Sigma$-measurable sets. 
Then: 
So $\mu N$ is countably additive, and we have $(2)$. 
=== Proof of $(3)$ ===
We have: 
So there exists $E \in \Sigma$ such that: 
:$\map {\paren {\mu N} } E \in \R$
verifying $(3)$.
== Sources ==
* : $\S 9$: Problem $11 \ \text{(i)}$
Category:Kernel Transformation of Measure
Category:Measures
Kernel Transformation of Positive Measurable Function is Positive Measurable Function

== Theorem ==
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.
Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel.
Let $f: X \to \overline{\R}$ be a positive measurable function.
Then $N f: X \to \overline{\R}$, the transformation of $f$ by $N$, is also a positive measurable function.
== Proof ==
== Sources ==
* : $\S 9$: Problem $11 \ \text{(ii)}$
Category:Measure Theory
Kernel is G-Module

== Theorem ==
Let $\struct {G, \cdot}$ be a group.
Let $f: \struct {V, \phi} \to \struct {V', \mu}$ be a homomorphism of $G$-modules.
Then its kernel $\map \ker f$ is a $G$-submodule of $V$.
== Proof ==
From G-Submodule Test it suffices to prove that $\phi \sqbrk {\struct {G, \map \ker f} } \subseteq \map \ker f$. 
That is, it is to be shown that, if $g \in G$ and $v \in \map \ker f$, then $\map \phi {g, v} \in \map \ker f$.
Assume that $g \in G$ and $v \in \map \ker f$.
Thus $\map \phi {g, v} \in \map \ker f$.
Hence $\map \ker f$ is a $G$-submodule of $V$.
Category:Representation Theory
Kernel is Normal Subgroup of Domain

== Theorem ==
Let $\phi$ be a group homomorphism.
Then the kernel of $\phi$ is a normal subgroup of the domain of $\phi$:
:$\map \ker \phi \lhd \Dom \phi$
== Proof ==
Let $\phi: G_1 \to G_2$ be a group homomorphism, where the identities of $G_1$ and $G_2$ are $e_{G_1}$ and $e_{G_2}$ respectively.
By Kernel of Group Homomorphism is Subgroup:
:$\map \ker \phi \le \Dom \phi$
Let $k \in \map \ker \phi, x \in G_1$.
Then:
So:
:$x k x^{-1} \in \map \ker \phi$
This is true for all $k \in \map \ker \phi$ and $x \in G_1$.
From Subgroup is Normal iff Contains Conjugate Elements, it follows that $\map \ker \phi$ is a normal subgroup of $G_1$.
== Also see ==
* Definition:Quotient Group Epimorphism
== Sources ==
* : Chapter $0$: Algebraic Concepts
* : $\S 7.4$. Kernel and image
* : Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Theorem $12.6$
* : $\S 1.10$: Theorem $22 \ (1)$
* : Chapter $\text{II}$: Groups: Morphisms
* : $\S 2.2$: Homomorphisms
* : Chapter $2$: Group Homomorphism and Isomorphism: $\S 65$
* : $\S 7$: Homomorphisms and quotient algebras
* : $\S 49.4$ Normal subgroups
* : Chapter $8$: The Homomorphism Theorem: Theorem $8.13: \ (1)$
* 
* : Chapter $13$: Corollary $13.20$
* 
Category:Kernels of Group Homomorphisms
Category:Normal Subgroups