nyuuzyou/wikis huggingface dataset:
Digamma Function of Two Thirds

== Theorem ==
:$\map \psi {\dfrac 2 3} = -\gamma - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$
where:
:$\psi$ denotes the digamma function
:$\gamma$ denotes the Euler-Mascheroni constant.
== Proof ==
Category:Examples of Digamma Function
Category:Euler-Mascheroni Constant
Digamma Reflection Formula

== Theorem ==
Let $z \in \C \setminus \Z$.
Let $\psi$ denote the digamma function.
Then:
:$\map \psi z - \map \psi {1 - z} = -\pi \cot \pi z$
== Proof ==
=== Lemma ===
Finally, from the , we note that:
:$\map \psi z - \map \psi {1 - z}$
is defined on the domain $\C \setminus \Z$.
== Also see ==
* Euler's Reflection Formula
* Polygamma Reflection Formula
== Sources == 
* 
Category:Digamma Function
Category:Reflection Formulas
Digital Root/Examples/34716 Base 10

== Example of Digital Root ==
In conventional base $10$ notation, we have:
:$\map {s_{10} } {34716} = 3 + 4 + 7 + 1 + 6 = 21$
and then:
:$\map {s_{10} } {21} = 2 + 1 = 3$.
So the digital root of $34716$ (base $10$) is $3$.
Category:Digital Roots
Digital Root of 3 Consecutive Numbers ending in Multiple of 3

== Theorem ==
Let $n$, $n + 1$ and $n + 2$ be positive integers such that $n + 2$ is a multiple of $3$.
Let $m = n + \paren {n + 1} + \paren {n + 2}$.
Then the digital root of $m$ is $6$.
== Proof ==
Let $n + 2$ be expressed as $3 r$ for some positive integer $r$.
Then:
The result follows from Digital Root of Number equals its Excess over Multiple of 9.
== Historical Note ==
== Sources ==
* : $6$
* : $6$
Category:6
Category:Digital Roots
Digital Root of Square

== Theorem ==
Let $n^2$ be a square number.
Then the digital root of $n^2$ is $1$, $4$, $7$ or $9$.
== Proof ==
Let $\map d n$ denote the digital root base $10$ of $n$.
From Digital Root is Congruent to Number Modulo Base minus 1, $\map d n \equiv n \pmod 9$.
So, let $n = 9 k + m$ where $1 \le m \le 9$.
Thus:
We enumerate the squares of the digits:
Hence the result.
Category:Digital Roots
Category:Square Numbers
Digits of Permutable Prime

== Theorem ==
Let $p$ be a permutable prime with more than $1$ digit.
Then $p$ contains only digits from the set:
:$\left\{ {1, 3, 7, 9}\right\}$
== Proof ==
First note that from 3-Digit Permutable Primes, both $337$ and $199$ are permutable primes.
Hence it follows that all the elements of $\left\{ {1, 3, 7, 9}\right\}$ appear in at least one permutable prime.
Let $p$ contain an even digit $d$.
Then at least one anagram $p'$ of $p$ has $d$ at the end.
From Divisibility by 2, it follows that $p'$ is even.
As $p$ has more than $1$ digit it follows that $p'$ is composite.
Let $p$ contain the digit $5$.
Then at least one anagram $p'$ of $p$ has $5$ at the end.
From Divisibility by 5, it follows that $p'$ is divisible by $5$.
Except for $5$ itself, all integers which are divisible by $5$ are composite.
As $p$ has more than $1$ digit it follows that $p'$ is composite.
The result follows by definition of permutable prime.
Category:Permutable Primes
Digraph/Examples/Arbitrary Digraph of Order 6

== Example of Digraph ==
Let $D = \struct {V, E}$ be a digraph such that:
:$V = \set {v_1, v_2, v_3, v_4, v_5, v_6}$
:$E = \set {\tuple {v_1, v_3}, \tuple {v_2, v_3}, \tuple {v_3, v_4}, \tuple {v_4, v_1}, \tuple {v_4, v_3}, \tuple {v_5, v_6} }$
Then $D$ can be presented in diagram form as:
:
== Sources ==
* : Chapter $1$: Mathematical Models: $\S 1.5$: Directed Graphs as Mathematical Models: Problem $32$
Category:Examples of Digraphs
Dihedral Group/Examples

== Examples of Dihedral Groups ==
=== Dihedral Group $D_1$ ===
=== Dihedral Group $D_2$ ===
=== Dihedral Group $D_3$ ===
=== Dihedral Group $D_4$ ===
=== Dihedral Group $D_5$ ===
=== Dihedral Group $D_6$ ===
Category:Examples of Dihedral Groups
Dihedral Group/Group Presentation

== Theorem ==
The dihedral group $D_n$ has the group presentation:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
That is, the dihedral group $D_n$ is generated by two elements $\alpha$ and $\beta$ such that:
:$(1): \quad \alpha^n = e$
:$(2): \quad \beta^2 = e$
:$(3): \quad \beta \alpha = \alpha^{n - 1} \beta$
== Proof ==
By definition, the dihedral group $D_n$ of order $2 n$ is the group of symmetries of the regular $n$-gon.
So, let $P$ denote a regular polygon with $n$ sides.
Let $\alpha$ be a rotation of $P$ by $\dfrac {2 \pi} n$.
Let $\beta$ be a reflection $P$ whose axis of reflection is the $y$ axis.
It takes $n$ rotations by $\dfrac {2 \pi} n$ to return $P$ to its original position.
Thus $\alpha$ is of order $n$, and so:
:$\alpha^n = e$
Two reflections of $P$ in the same axis of reflection also returns $P$ to its original position.
Thus $\beta$ is of order $2$, and so:
:$\beta^2 = e$
It remains to be shown that $\beta \alpha \beta = \alpha^{-1}$.
Let the vertices of $P$ be labelled in order anticlockwise from the $y$-axis: $A_1, A_2, \ldots, A_n$.
Let $P$ be arranged so that $A_1$ is on the $y$-axis.
From here, $\beta$ on $P$ exchanges the places of $A_2$ and $A_n$.
From here, $\alpha$ sends $A_2$ to the position of $A_1$, $A_1$ to the original position of $A_2$, and $A_3$ to the original position of $A_n$.
From here, $\beta$ sends exchanges the positions of $A_3$ and $A_1$.
So $\beta \alpha \beta$ on $P$ results in $A_1$ moving to the original position of $A_n$, $A_2$ moving to the original position of $A_1$ and $A_3$ moving to the original position of $A_2$.
That is, $\beta \alpha \beta$ is a rotation of $P$ by $-\dfrac {2 \pi} n$, or $\alpha^{-1}$.
== Sources ==
* : Chapter $2$: Subgroups and Cosets: $\S 35 \eta$
* : Chapter $4$: Subgroups: Example $4.10$
* 
* 
Category:Dihedral Groups
Category:Examples of Group Presentations
Dihedral Group D3/Cayley Table

== Cayley Table of Dihedral Group $D_3$ ==
The Cayley table of the dihedral group $D_3$ can be written:
:$\begin{array}{c|cccccc} & e & a & a^2 & b & a b & a^2 b \\ \hline e & e & a & a^2 & b & a b & a^2 b \\ a & a & a^2 & e & a b & a^2 b & b \\ a^2 & a^2 & e & a & a^2 b & b & a b \\ b & b & a^2 b & a b & e & a^2 & a \\ a b & a b & b & a^2 b & a & e & a^2 \\ a^2 b & a^2 b & a b & b & a^2 & a & e \\ \end{array}$
where the group presentation of $D_3$ is given as:
Category:Examples of Cayley Tables
Category:Dihedral Group D3
Dihedral Group D3/Group Presentation

== Group Presentation of Dihedral Group $D_3$ ==
The group presentation of the dihedral group $D_3$ is given by:
:$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$
== Proof ==
We have that the group presentation of the dihedral group $D_n$ is:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Setting $n = 3, \alpha = a, \beta = b$, we get:
:$D_3 = \gen {a, b: a^3 = b^2 = e, b a b = a^{−1} }$
from which the result follows.
Category:Dihedral Group D3
Dihedral Group D4/Cayley Table

== Cayley Table for Dihedral Group $D_4$ ==
The Cayley table for the dihedral group $D_4$, whose group presentation is:
can be presented as:
:$\begin{array}{l|cccccccc} & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ \hline e & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ a & a & a^2 & a^3 & e & b a^3 & b & b a & b a^2 \\ a^2 & a^2 & a^3 & e & a & b a^2 & b a^3 & b & b a \\ a^3 & a^3 & e & a & a^2 & b a & b a^2 & b a^3 & b \\ b & b & b a & b a^2 & b a^3 & e & a & a^2 & a^3 \\ b a & b a & b a^2 & b a^3 & b & a^3 & e & a & a^2 \\ b a^2 & b a^2 & b a^3 & b & b a & a^2 & a^3 & e & a \\ b a^3 & b a^3 & b & b a & b a^2 & a & a^2 & a^3 & e \end{array}$
=== Coset Decomposition of $\set {e, a^2}$ ===
Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\gen{a^2}$ gives:
=== Coset Decomposition of $\set {e, a, a^2, a^3}$ ===
Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\gen a$ gives:
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Example $7.13$
Category:Dihedral Group D4
Category:Examples of Cayley Tables
Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a, a^2, a^3)

== Cayley Table for Dihedral Group $D_4$ ==
The Cayley table for the dihedral group $D_4$, with respect to the coset decomposition of the normal subgroup $\gen a$, is:
can be presented as:
:$\begin{array}{l|cccc|cccc} & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ \hline e & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ a & a & a^2 & a^3 & e & b a^3 & b & b a & b a^2 \\ a^2 & a^2 & a^3 & e & a & b a^2 & b a^3 & b & b a \\ a^3 & a^3 & e & a & a^2 & b a & b a^2 & b a^3 & b \\ \hline b & b & b a & b a^2 & b a^3 & e & a & a^2 & a^3 \\ b a & b a & b a^2 & b a^3 & b & a^3 & e & a & a^2 \\ b a^2 & b a^2 & b a^3 & b & b a & a^2 & a^3 & e & a \\ b a^3 & b a^3 & b & b a & b a^2 & a & a^2 & a^3 & e \end{array}$
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Example $7.13$
Category:Dihedral Group D4
Category:Examples of Cayley Tables
Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a^2)

== Cayley Table for Dihedral Group $D_4$ ==
The Cayley table for the dihedral group $D_4$, with respect to the coset decomposition of the normal subgroup $\gen {a^2}$, is:
can be presented as:
:$\begin{array}{l|cc|cc|cc|cc} & e & a^2 & a & a^3 & b & b a^2 & b a & b a^3 \\ \hline e & e & a^2 & a & a^3 & b & b a^2 & b a & b a^3 \\ a^2 & a^2 & e & a^3 & a & b a^2 & b & b a^3 & b a \\ \hline a & a & a^3 & a^2 & e & b a^3 & b a & b & b a^2 \\ a^3 & a^3 & a & e & a^2 & b a & b a^3 & b a^2 & b \\ \hline b & b & b a^2 & b a & b a^3 & e & a^2 & a & a^3 \\ b a^2 & b a^2 & b & b a^3 & b a & a^2 & e & a^3 & a \\ \hline b a & b a & b a^3 & b a^2 & b & a^3 & a & e & a^2 \\ b a^3 & b a^3 & b a & b & b a^2 & a & a^3 & a^2 & e \end{array}$
which is seen to be an example of the Klein $4$-group.
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Example $7.13$
Category:Dihedral Group D4
Category:Examples of Cayley Tables
Dihedral Group D4/Center

== Center of the Dihedral Group $D_4$ ==
Let $D_4$ denote the dihedral group $D_4$, whose group presentation is given as:
The center of $D_4$ is given by:
:$\map Z {D_4} = \set {e, a^2}$
== Proof ==
From Center of Dihedral Group:
:$\map Z {D_n} = \begin{cases} e & : n \text { odd} \\ \set {e, \alpha^{n / 2} } & : n \text { even} \end{cases}$
Hence the result.
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Exercise $5$
Category:Centers of Groups
Category:Dihedral Group D4
Dihedral Group D4/Group Presentation

== Group Presentation of Dihedral Group $D_4$ ==
The group presentation of the dihedral group $D_4$ is given by:
:$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$
== Proof ==
We have that the group presentation of the dihedral group $D_n$ is:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Setting $n = 4, \alpha = a, \beta = b$, we get:
:$D_4 = \gen {a, b: a^4 = b^2 = e, b a b = a^{−1} }$
from which the result follows.
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 1

== Matrix Representation of Dihedral Group $D_4$ ==
Let $\mathbf I, \mathbf A, \mathbf B, \mathbf C$ denote the following four elements of the matrix space $\map {\MM_\Z} 2$:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
The set:
:$D_4 = \set {\mathbf I, -\mathbf I, \mathbf A, -\mathbf A, \mathbf B, -\mathbf B, \mathbf C, -\mathbf C}$
under the operation of conventional matrix multiplication, forms the dihedral group $D_4$.
=== Cayley Table ===
Its Cayley table is given by:
== Also see ==
* Dihedral Group D4 Defined by Matrices where it is shown that these have the appropriate properties.
== Sources ==
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table

== Cayley Table for Dihedral Group $D_4$ ==
The Cayley table for the dihedral group $D_4$:
:$D_4 = \set {\mathbf I, -\mathbf I, \mathbf A, -\mathbf A, \mathbf B, -\mathbf B, \mathbf C, -\mathbf C}$
under the operation of conventional matrix multiplication, where:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
can be presented as:
:$\begin{array}{r|rrrrrrrr}
 & \mathbf I & \mathbf A & \mathbf B & \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C \\
\hline
 \mathbf I & \mathbf I & \mathbf A & \mathbf B & \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C \\
 \mathbf A & \mathbf A & \mathbf I & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I & \mathbf C & \mathbf B \\
 \mathbf B & \mathbf B & \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C & \mathbf I & \mathbf A \\
 \mathbf C & \mathbf C & \mathbf B & \mathbf A & \mathbf I & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I \\
-\mathbf I & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C & \mathbf I & \mathbf A & \mathbf B & \mathbf C \\
-\mathbf A & -\mathbf A & -\mathbf I & \mathbf C & \mathbf B & \mathbf A & \mathbf I & -\mathbf C & -\mathbf B \\
-\mathbf B & -\mathbf B & -\mathbf C & \mathbf I & \mathbf A & \mathbf B & -\mathbf C & -\mathbf I & -\mathbf A \\
-\mathbf C & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I & \mathbf C & \mathbf B & \mathbf A & \mathbf I
\end{array}$
== Sources ==
Category:Dihedral Group D4
Category:Examples of Cayley Tables
Dihedral Group D4/Matrix Representation/Formulation 2

== Matrix Representation of Dihedral Group $D_4$ ==
Let $\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G$ denote the following $8$ elements of the matrix space $\map {\MM_\Z} 2$:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
The set:
:$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$
under the operation of conventional matrix multiplication, forms the dihedral group $D_4$.
=== Cayley Table ===
Its Cayley table is given by:
== Generated Subgroups ==
== Cosets ==
== Also see ==
* Dihedral Group D4 Defined by Matrices where it is shown that these have the appropriate properties.
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Cayley Table

== Cayley Table for Dihedral Group $D_4$ ==
The Cayley table for the dihedral group $D_4$:
:$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$
under the operation of conventional matrix multiplication, where:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
can be presented as:
:$\begin{array}{r|rrrrrrrr}
 & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\
\hline
\mathbf I & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\
\mathbf A & \mathbf A & \mathbf B & \mathbf C & \mathbf I & \mathbf E & \mathbf F & \mathbf G & \mathbf D \\
\mathbf B & \mathbf B & \mathbf C & \mathbf I & \mathbf A & \mathbf F & \mathbf G & \mathbf D & \mathbf E \\
\mathbf C & \mathbf C & \mathbf I & \mathbf A & \mathbf B & \mathbf G & \mathbf D & \mathbf E & \mathbf F \\
\mathbf D & \mathbf D & \mathbf G & \mathbf F & \mathbf E & \mathbf I & \mathbf C & \mathbf B & \mathbf A \\
\mathbf E & \mathbf E & \mathbf D & \mathbf G & \mathbf F & \mathbf A & \mathbf I & \mathbf C & \mathbf B \\
\mathbf F & \mathbf F & \mathbf E & \mathbf D & \mathbf G & \mathbf B & \mathbf A & \mathbf I & \mathbf C \\
\mathbf G & \mathbf G & \mathbf F & \mathbf E & \mathbf D & \mathbf C & \mathbf B & \mathbf A & \mathbf I
\end{array}$
== Sources ==
* : Chapter $1$: Definitions and Examples: Exercise $2$
Category:Dihedral Group D4
Category:Examples of Cayley Tables
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets

== Examples of Cosets of Subgroups of Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by the set of square matrices:
:$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$
under the operation of conventional matrix multiplication, where:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
=== Generated Subgroup $\gen {\mathbf A}$ ===
=== Generated Subgroup $\gen {\mathbf B, \mathbf F}$ ===
=== Generated Subgroup $\gen {\mathbf D}$ ===
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by A

== Examples of Cosets of Subgroups of Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by the set of square matrices:
:$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$
under the operation of conventional matrix multiplication, where:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
Let $H \subseteq D_4$ be defined as:
:$H = \gen {\mathbf A}$
where $\gen {\mathbf A}$ denotes the subgroup generated by $\mathbf A$.
From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf A}$ we have:
:$\gen {\mathbf A} = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}$
=== Left Cosets ===
=== Right Cosets ===
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by A/Left Cosets

== Examples of Left Cosets of Subgroups of Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by the set of square matrices:
:$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$
under the operation of conventional matrix multiplication, where:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
Let $H \subseteq D_4$ be defined as:
:$H = \gen {\mathbf A}$
where $\gen {\mathbf A}$ denotes the subgroup generated by $\mathbf A$.
From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf A}$ we have:
:$\gen {\mathbf A} = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}$
The left cosets of $H$ are:
== Proof ==
The Cayley table of $D_4$ is presented as:
Thus:
</onlyinclude>
== Sources ==
* : Chapter $5$: Cosets and Lagrange's Theorem: Exercise $1 \ \text {(a)}$
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by A/Right Cosets

== Examples of Right Cosets of Subgroups of Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by the set of square matrices:
:$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$
under the operation of conventional matrix multiplication, where:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
Let $H \subseteq D_4$ be defined as:
:$H = \gen {\mathbf A}$
where $\gen {\mathbf A}$ denotes the subgroup generated by $\mathbf A$.
From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf A}$ we have:
:$\gen {\mathbf A} = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}$
The right cosets of $H$ are:
== Proof ==
The Cayley table of $D_4$ is presented as:
Thus:
</onlyinclude>
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by B, F

== Examples of Cosets of Subgroups of Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by the set of square matrices:
:$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$
under the operation of conventional matrix multiplication, where:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
Let $H \subseteq D_4$ be defined as:
:$H = \gen {\mathbf B, \mathbf F}$
where $\gen {\mathbf B, \mathbf F}$ denotes the subgroup generated by $\set {\mathbf B, \mathbf F}$.
From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf B, \mathbf F}$ we have:
:$\gen {\mathbf B, \mathbf F} = \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$
=== Left Cosets ===
=== Right Cosets ===
Category:Examples of Cosets
Category:Dihedral Group D4