nyuuzyou/wikis huggingface dataset:
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by B, F/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:
:$H = \gen {\mathbf B, \mathbf F}$
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:
:$\gen {\mathbf B, \mathbf F} = \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$
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 $1 \ \text {(b)}$
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by B, F/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:
:$H = \gen {\mathbf B, \mathbf F}$
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:
:$\gen {\mathbf B, \mathbf F} = \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$
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 D

== 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:
:$H = \gen {\mathbf D}$
where $\gen {\mathbf D}$ denotes the subgroup generated by $\mathbf D$.
From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf D}$ we have:
:$\gen {\mathbf D} = \set {\mathbf I, \mathbf D}$
=== 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 D/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:
:$H = \gen {\mathbf D}$
where $\gen {\mathbf D}$ denotes the subgroup generated by $\mathbf D$.
From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf D}$ we have:
:$\gen {\mathbf D} = \set {\mathbf I, \mathbf D}$
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 $1 \ \text {(c)}$
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by D/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:
:$H = \gen {\mathbf D}$
where $\gen {\mathbf D}$ denotes the subgroup generated by $\mathbf D$.
From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf D}$ we have:
:$\gen {\mathbf D} = \set {\mathbf I, \mathbf D}$
The right 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 $1 \ \text {(c)}$
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups

== Examples of Generated 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 $\gen {\mathbf A, \mathbf D}$ ===
=== Generated Subgroup $\gen {\mathbf A, \mathbf C}$ ===
=== Generated Subgroup $\gen {\mathbf D}$ ===
=== Generated Subgroup $\gen {\mathbf B, \mathbf F}$ ===
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups/A

== Examples of Generated 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:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
The generated subgroup $\gen {\mathbf A}$ is the cyclic group of order $4$:
:$\set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}$
== Proof ==
The Cayley table of $D_4$ is as follows:
We have:
Thus:
:$\gen {\mathbf A} = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}$
That this is a cyclic group follows by definition.
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups/A, C

== Examples of Generated 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:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
The generated subgroup $\gen {\mathbf A, \mathbf C}$ is the cyclic group of order $4$:
:$\set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}$
== Proof ==
The Cayley table of $D_4$ is as follows:
We have:
Thus:
:$\gen {\mathbf A, \mathbf D} = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}$
That this is a cyclic group follows by definition, and that:
:$\gen {\mathbf A, \mathbf D} = \gen {\mathbf A}$
== Sources ==
* : Chapter $4$: Subgroups: Exercise $3$
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups/A, D

== Examples of Generated 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:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
The generated subgroup $\gen {\mathbf A, \mathbf D}$ is the whole of $D_4$.
== Proof ==
The Cayley table of $D_4$ is as follows:
We have:
Thus for a start $\mathbf I, \mathbf A, \mathbf B, \mathbf C \in \gen {\mathbf A, \mathbf D}$.
Then:
and so it follows that $\gen {\mathbf A, \mathbf D} = D_4$.
== Sources ==
* : Chapter $4$: Subgroups: Exercise $3$
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups/B, F

== Examples of Generated 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:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
The generated subgroup $\gen {\mathbf D, \mathbf F}$ is the Klein $4$-group:
:$\set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$
== Proof ==
The Cayley table of $D_4$ is as follows:
We have:
Thus:
:$\gen {\mathbf B, \mathbf D} = \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$
and its Cayley table is:
:$\begin{array}{r|rrrr}
 & \mathbf I & \mathbf B & \mathbf D & \mathbf F \\
\hline
\mathbf I & \mathbf I & \mathbf B & \mathbf D & \mathbf F \\
\mathbf B & \mathbf B & \mathbf I & \mathbf F & \mathbf D \\
\mathbf D & \mathbf D & \mathbf F & \mathbf I & \mathbf B \\
\mathbf F & \mathbf F & \mathbf D & \mathbf B & \mathbf I \\
\end{array}$
This is seen to be the Klein $4$-group.
== Sources ==
* : Chapter $4$: Subgroups: Exercise $3$
Category:Dihedral Group D4
Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups/D

== Examples of Generated 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:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad
\mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}
\qquad
\mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\qquad
\mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
:$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\qquad
\mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}
\qquad
\mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
The generated subgroup $\gen {\mathbf D}$ is the (cyclic) group of order $2$:
:$\set {\mathbf I, \mathbf D}$
== Proof ==
The Cayley table of $D_4$ is as follows:
We have:
Thus:
:$\gen {\mathbf D} = \set {\mathbf I, \mathbf D}$
and its Cayley table is:
:$\begin{array}{r|rr}
 & \mathbf I & \mathbf D \\
\hline
\mathbf I & \mathbf I & \mathbf D \\
\mathbf D & \mathbf D & \mathbf I \\
\end{array}$
This is seen to be the (cyclic) group of order $2$.
Category:Dihedral Group D4
Dihedral Group D4/Normal Subgroups/Subgroup Generated by a

== Example of Normal Subgroup of the Dihedral Group $D_4$==
Let the dihedral group $D_4$ be represented by its group presentation:
The subgroup of $D_4$ generated by $\gen a$ is normal.
== Proof ==
Let $N = \gen a$
First it is noted that as $a^4 = e$:
:$N = \set {e, a, a^2, a^3}$
and is cyclic.
The left cosets of $N$:
As $\order {\gen a} = \order {D_4} / 2$ it follows from Subgroup of Index 2 is Normal that $\gen a$ is normal.
== Sources ==
* : Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures
* : Chapter $7$: Normal subgroups and quotient groups: Example $7.13$
Category:Dihedral Group D4
Category:Examples of Normal Subgroups
Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2

== Example of Normal Subgroup of the Dihedral Group $D_4$==
Let the dihedral group $D_4$ be represented by its group presentation:
The subgroup of $D_4$ generated by $\gen {a^2}$ is normal.
=== Quotient Group ===
== Proof ==
Let $N = \gen {a^2}$
First it is noted that as $\paren {a^2}^2 = a^4 = e$:
:$N = \set {e, a^2}$
The left cosets of $N$:
As $b a^2 = a^2 b$ and $b a^3 = a^2 b a$, it follows immediately that:
:$b N = N b, a N = N a, b a N = N b a$
and so $\gen {a^2}$ is seen to be normal.
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Example $7.13$
Category:Dihedral Group D4
Category:Examples of Normal Subgroups
Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2, b

== Example of Normal Subgroup of the Dihedral Group $D_4$==
Let the dihedral group $D_4$ be represented by its group presentation:
The subgroup of $D_4$ generated by $\gen {a^2, b}$ is normal.
== Proof ==
Let $N = \gen {a^2, b}$.
First it is noted that:
so:
:$N = \set {e, a^2, b, b a^2}$
and is an instance of the Klein $4$-group.
The left cosets of $N$:
As $\order {\gen {a^2, b} } = \order {D_4} / 2$ it follows from Subgroup of Index 2 is Normal that $\gen {a^2, b}$ is normal.
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Example $7.15$
Category:Dihedral Group D4
Category:Examples of Normal Subgroups
Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2/Quotient Group

== Quotient Group of Normal Subgroup of the Dihedral Group $D_4$==
Let the dihedral group $D_4$ be represented by its group presentation:
Consider the normal subgroup $\gen {a^2}$ of $D_4$:
:$\gen {a^2} = \set {e, a^2}$
Let $E := N, A := a N, B := b N, C := a b N$.
Thus the quotient group of $G$ by $N$ is:
:$G / N = \set {E, A, B, C}$
whose Cayley table can be presented as:
:$\begin{array}{c|cccc}
 & E & A & B & C \\
\hline
E & E & A & B & C \\
A & A & E & C & B \\
B & B & C & E & A \\
C & C & B & A & E \\
\end{array}$
which is seen to be an example of the Klein $4$-group.
=== Subgroups of Quotient ===
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Example $7.13$
Category:Dihedral Group D4
Category:Examples of Normal Subgroups
Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2/Quotient Group/Subgroups

== Quotient Group of Normal Subgroup of the Dihedral Group $D_4$==
Let the dihedral group $D_4$ be represented by its group presentation:
Consider the quotient group of the normal subgroup $\gen {a^2}$ of $D_4$:
:$G / N = \set {E, A, B, C}$
where:
:$E := \set {e, a^2}, A := a \set {a, a^3}, B := b \set {b, b a^2}, C := a b \set {b a, b a^3}$.
The subgroups of $G / N$ are:
As $G / N$ is abelian, all of these subgroups are normal in $G / N$.
Also, all of these are normal in $D_4$:
:$E$ is normal 
:$\set {E, A}, \set {E, B}, \set {E, C}$ are all of index $2$, and so normal by Subgroup of Index 2 is Normal.
:$D_4$ is normal by Group is Normal in Itself.
.
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Example $7.15$
Category:Dihedral Group D4
Category:Examples of Normal Subgroups
Dihedral Group D4/Subgroups

== Subgroups of the Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by its group presentation:
The subsets of $D_4$ which form subgroups of $D_4$ are:
== Proof ==
Consider the Cayley table for $D_4$:
We have that:
:$a^4 = e$
and so $\gen a = \set {e, a, a^2, a^3}$ forms a subgroup of $D_4$ which is cyclic.
We have that:
:$\paren {a^2}^2 = e$
and so $\gen {a^2} = \set {e, a^2}$ forms a subgroup of $D_4$ which is cyclic, and also a subgroup of $\gen a$.
We have that:
:$b^2 = e$
and so $\gen b = \set {e, b}$ forms a subgroup of $D_4$ which is cyclic.
We have that:
:$\paren {b a}^2 = e$
and so $\gen {b a} = \set {e, b a}$ forms a subgroup of $D_4$ which is cyclic.
We have that:
:$\paren {b a^2}^2 = e$
and so $\gen {b a^2} = \set {e, b a^2}$ forms a subgroup of $D_4$ which is cyclic.
We have that:
:$\paren {b a^3}^2 = e$
and so $\gen {b a^3} = \set {e, b a^3}$ forms a subgroup of $D_4$ which is cyclic.
Then we have that:
:$b a^2 = a^2 b$
and so $\gen {a^2, b} = \set {e, a^2, b, b a^2}$ forms a subgroup of $D_4$ which is not cyclic, but which has subgroups $\set {e, a^2}$, $\set {e, b}$, $\set {e, b a^2}$.
Then we have that:
:$b a^3 = a^2 b a$
and so $\gen {a^2, b a} = \set {e, a^2, b a, b a^3}$ forms a subgroup of $D_4$ which is not cyclic, but which has subgroups $\set {e, a^2}$, $\set {e, b}$, $\set {e, b a^2}$.
That exhausts all elements of $D_4$.
Any subgroup generated by any $2$ elements of $Q$ which are not both in the same subgroup as described above generate the whole of $D^4$.
Category:Dihedral Group D4
Dihedral Group D4/Subgroups/Cosets/Subgroup Generated by b

== Examples of Cosets of Subgroups of Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by its group presentation:
Let $H \subseteq D_4$ be defined as:
:$H = \gen b$
where $\gen b$ denotes the subgroup generated by $b$.
From Subgroups of Dihedral Group D4 we have:
:$\gen b = \set {e, b}$
=== Left Cosets ===
=== Right Cosets ===
It is seen that the left cosets do not equal the corresponding right cosets.
It follows by definition that $\gen b$ is not a normal subgroup of $D_4$.
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Exercise $4$
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4/Subgroups/Cosets/Subgroup Generated by b/Left Cosets

== Examples of Left Cosets of Subgroups of Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by its group presentation:
Let $H \subseteq D_4$ be defined as:
:$H = \gen b$
where $\gen b$ denotes the subgroup generated by $b$.
From Subgroups of Dihedral Group D4 we have:
:$\gen b = \set {e, b}$
The left cosets of $H$ are:
== Proof ==
The Cayley table of $D_4$ is presented as:
Thus:
</onlyinclude>
== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Exercise $4$
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4/Subgroups/Cosets/Subgroup Generated by b/Right Cosets

== Examples of Left Cosets of Subgroups of Dihedral Group $D_4$ ==
Let the dihedral group $D_4$ be represented by its group presentation:
Let $H \subseteq D_4$ be defined as:
:$H = \gen b$
where $\gen b$ denotes the subgroup generated by $b$.
From Subgroups of Dihedral Group D4 we have:
:$\gen b = \set {e, b}$
The right cosets of $H$ are:
== Proof ==
The Cayley table of $D_4$ is presented as:
Thus:
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== Sources ==
* : Chapter $7$: Normal subgroups and quotient groups: Exercise $4$
Category:Examples of Cosets
Category:Dihedral Group D4
Dihedral Group D4 is not Internal Group Product/Proof 1

== Theorem ==
== Proof ==
 $D_4$ is the internal group product of $2$ proper subgroups $H$ and $K$ of $D_4$.
, let $\order H = 2$ and $\order K = 4$.
$H$ needs to be normal in $D_4$ for the conditions of the internal group product to be satisfied.
So the non-identity element of $H$ needs to be conjugated to itself by every element of $G$.
This means $H$ is a subset of the center of $G$.
From Center of Dihedral Group $D_4$:
:$H = \set {e, a^2}$
But every non-trivial normal subgroup of $D_4$ contains $a^2$.
This makes it impossible for $H \cap K = \set e$.
Thus $D_4$ cannot be an internal group product.
== Sources ==
* : Chapter $13$: Direct products: Exercise $1$
Category:Dihedral Group D4 is not Internal Group Product
Dihedral Group D4 is not Internal Group Product/Proof 2

== Theorem ==
== Proof ==
The proper subgroups of $D_4$ are of order no greater than $4$.
From Group of Order less than 6 is Abelian, all such proper subgroups are abelian.
From External Direct Product of Abelian Groups is Abelian Group, the group direct product of $2$ of these proper subgroups is in turn abelian.
From Internal and External Group Direct Products are Isomorphic, the internal group product of $2$ of these proper subgroups is in turn abelian.
But $D_4$ is not abelian.
Hence it cannot be the internal group product of any $2$ of these proper subgroups.
Category:Dihedral Group D4 is not Internal Group Product