View source for Element structure of dihedral group:D8

{{group-specific information|
information type = element structure|
group = dihedral group:D8|
connective = of}}

We denote the identity element by <math>e</math>. The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or the dihedral group acting on four elements, is defined by the following presentation:

<math>\langle x,a| a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle</math>

<section begin="multiplication table"/>

The row element is multiplied on the left and the column element is multiplied on the right.

{| class="sortable" border="1"
!Element !! <math>\! e</math> !! <math>\! a</math> !! <math>\! a^2</math> !! <math>\! a^3</math> !! <math>\! x</math> !! <math>\! ax</math> !! <math>\! a^2x</math> !! <math>\! a^3x</math>
|-
| <math>\! e</math> || <math>\! e</math> || <math>\! a</math> || <math>\! a^2</math> || <math>\! a^3</math> || <math>\! x</math> || <math>\! ax</math> || <math>\! a^2x</math> || <math>\! a^3x</math>
|-
| <math>\! a</math> || <math>\! a</math> ||<math>\! a^2</math> || <math>\! a^3</math> ||<math>\! e</math>  || <math>\! ax</math> || <math>\! a^2x</math> || <math>\! a^3x</math> || <math>\! x</math>
|-
| <math>\! a^2</math> || <math>\! a^2</math> || <math>\! a^3</math> ||<math>\! e</math>  ||<math>\! a</math> ||<math>\! a^2x</math> || <math>\! a^3x</math> || <math>\! x</math> || <math>\! ax</math>
|-
| <math>\! a^3</math> || <math>\! a^3</math> ||<math>\! e</math>  ||<math>\! a</math> ||<math>\! a^2</math> ||<math>\! a^3x</math> || <math>\! x</math> || <math>\! ax</math> || <math>\! a^2x</math>
|-
| <math>\! x</math> || <math>\! x</math> || <math>\! a^3x</math> || <math>\! a^2x</math> || <math>\! ax</math> || <math>\! e</math> || <math>\! a^3</math> || <math>\! a^2</math> || <math>\! a</math>
|-
| <math>\! ax</math> || <math>\! ax</math> || <math>\! x</math> || <math>\! a^3x</math> || <math>\! a^2x</math> || <math>\! a</math>|| <math>\! e</math> || <math>\! a^3</math> || <math>\! a^2</math>
|-
| <math>\! a^2x</math> || <math>\! a^2x</math> || <math>\! ax</math> || <math>\! x</math> || <math>\! a^3x</math> || <math>\! a^2</math> || <math>\! a</math> || <math>\! e</math> || <math>\! a^3</math>
|-
| <math>\! a^3x</math> || <math>\! a^3x</math> || <math>\! a^2x</math> || <math>\! ax</math> || <math>\! x</math> || <math>\! a^3</math> || <math>\! a^2</math> || <math>\! a</math> || <math>\! e</math>
|}

<section end="multiplication table"/>
{{quotation|'''CAUTION''': You may be looking instead for [[dihedral group:D16]], the dihedral group of ''degree'' 8 and order 16. If so, see [[element structure of dihedral group:D16]].}}

==Summary==

{| class="sortable" border="1"
! Item !! Value
|-
| [[order of a group|order]] of the whole group (total number of elements) || 8
|-
| [[conjugacy class size statistics of a finite group|conjugacy class sizes]] || 1,1,2,2,2<br>[[maximum conjugacy class size|maximum]]: 2, [[number of conjugacy classes]]: 5, [[lcm of conjugacy class sizes|lcm]]: 2
|-
| [[order statistics of a finite group|order statistics]] || 1 of order 1, 5 of order 2, 2 of order 4<br>[[maximum of element orders|maximum]]: 4, [[exponent of a group|lcm (exponent of the whole group)]]: 4
|}

==Elements==

<section begin="elements"/>

Below, we list all the elements, also giving the interpretation of each element under the geometric description of the dihedral group as the symmetries of a 4-gon, and for the corresponding permutation representation (see [[D8 in S4]]). Note that for different conventions, one can obtain somewhat different correspondences, so this may not match up with other correspondences elsewhere. ''Note that the descriptions below assume the left action convention for functions and the corresponding convention for composition, and hence some of the entries may become different if you adopt the right action convention.'':

{| class="sortable" border="1"
! Element in terms of <math>a</math> and <math>x</math> !! Geometric description !! Permutation on vertices !! Order of the element
|-
| <math>e</math> (identity element) || does nothing, i.e., leaves the square invariant || <math>()</math> || 1
|-
| <math>a</math> || rotation by angle of <math>\pi/2</math> (i.e., <math>90\,^\circ</math>) counterclockwise || <math>(1,2,3,4)</math> || 4
|-
| <math>a^2</math> || rotation by angle of <math>\pi</math> (i.e., <math>180\,^\circ</math>), also called a ''half turn'' || <math>(1,3)(2,4)</math> || 2
|-
| <math>a^3</math> || rotation by angle of <math>3\pi/2</math> (i.e., <math>270\,^\circ</math>) counter-clockwise, or equivalently, by <math>\pi/2</math> (i.e., <math>90\,^\circ</math>) clockwise || <math>(1,4,3,2)</math> || 4
|-
| <math>x</math> || reflection about the diagonal joining vertices "2" and "4" || <math>(1,3)</math> || 2
|-
| <math>ax = xa^3</math> || reflection about the line joining midpoints of opposite sides "14" and "23" || <math>(1,4)(2,3)</math> || 2
|-
| <math>a^2x</math> || reflection about the diagonal joining vertices "1" and "3" || <math>(2,4)</math> || 2
|-
| <math>a^3x = xa</math> || reflection about the line joining midpoints of opposite sides "12" and "34" || <math>(1,2)(3,4)</math> || 2
|}
<section end="elements"/>

==Commutator map==

Because of the fact that the [[inner automorphism group]] is an elementary abelian 2-group, it does not matter which of the two definitions of commutator map we choose (<math>[\alpha,\beta] = \alpha^{-1}\beta^{-1}\alpha\beta</math> or <math>[\alpha,\beta] = \alpha\beta\alpha^{-1}\beta^{-1}</math>) -- they are both the same map.

In fact, the commutator map sends a pair of elements to <math>e</math> if they commute and to <math>a^2</math> if they don't commute.
{| class="sortable" border="1"
!Element !! <math>\! e</math> !! <math>\! a</math> !! <math>\! a^2</math> !! <math>\! a^3</math> !! <math>\! x</math> !! <math>\! ax</math> !! <math>\! a^2x</math> !! <math>\! a^3x</math>
|-
| <math>\! e</math> || <math>\! e</math> || <math>\! e </math> || <math>\! e</math> || <math>\! e </math> || <math>\! e</math> || <math>\! e </math> || <math>\! e</math> || <math>\! e </math> 
|-
| <math>\! a</math> || <math>\! e</math> || <math>\! e</math> || <math>\! e</math> || <math>\! e</math> || <math>\! a^2</math> || <math>\! a^2</math> || <math>\! a^2</math> || <math>\! a^2</math>
|-
| <math>\! a^2</math> || <math>\! e</math> || <math>\! e </math> || <math>\! e</math> || <math>\! e </math> || <math>\! e</math> || <math>\! e </math> || <math>\! e</math> || <math>\! e </math>
|-
| <math>\! a^3</math> || <math>\! e</math> || <math>\! e</math> || <math>\! e</math> || <math>\! e</math> || <math>\! a^2</math> || <math>\! a^2</math> || <math>\! a^2</math> || <math>\! a^2</math>
|-
| <math>\! x</math> || <math>\! e</math> || <math>\! a^2</math>|| <math>\! e</math> || <math>\! a^2</math>||<math>\! e</math> || <math>\! a^2</math>||<math>\! e</math> || <math>\! a^2</math>
|-
| <math>\! ax</math> || <math>\! e</math> || <math>\! a^2</math> || <math>\! e</math> || <math>\! a^2</math> || <math>\! a^2</math>|| <math>\! e</math> || <math>\! a^2</math> || <math>\! e</math>
|-
| <math>\! a^2x</math> || <math>\! e</math> || <math>\! a^2</math>|| <math>\! e</math> || <math>\! a^2</math>||<math>\! e</math> || <math>\! a^2</math>||<math>\! e</math> || <math>\! a^2</math>
|-
| <math>\! a^3x</math> || <math>\! e</math> || <math>\! a^2</math> || <math>\! e</math> || <math>\! a^2</math> || <math>\! a^2</math>|| <math>\! e</math> || <math>\! a^2</math> || <math>\! e</math>
|}


==Conjugacy class structure==

{{conjugacy class structure facts to check against}}

===General description===

<section begin="conjugacy and automorphism class structure"/>
{| class="sortable" border="1"
! Conjugacy class in terms of <math>a,x</math> !! Geometric description of conjugacy class !! Conjugacy class as permutations !! Size of conjugacy class !! Order of elements in conjugacy class !! Centralizer of first element of class
|-
| <math>\! \{ e \}</math> || identity element, does nothing || <math>\{ () \}</math>|| 1 || 1 || whole group
|-
| <math>\! \{ a^2 \}</math> || half turn, rotation by <math>\pi</math> || <math>\{ (1,3)(2,4) \}</math> || 1 || 2 || whole group
|-
| <math>\! \{ x,a^2x \}</math> || reflections about diagonals || <math>\{ (1,3), (2,4) \}</math> || 2 || 2 || <math>\{ e, a^2, x, a^2x \}</math> -- one of the [[Klein four-subgroups of dihedral group:D8]]
|-
| <math>\! \{ ax, a^3x \}</math> || reflections about lines joining midpoints of opposite sides || <math>\{ (1,4)(2,3)\ , \ (1,2)(3,4) \}</math> || 2 || 2 || <math>\{ e, a^2, ax, a^3x \}</math> -- one of the [[Klein four-subgroups of dihedral group:D8]]
|-
| <math>\! \{ a, a^3 \}</math> || rotations by odd multiples of <math>\pi/2</math> || <math>\{ (1,2,3,4) \ ,\  (1,4,3,2) \}</math> || 2 || 4 ||<math>\{ e, a, a^2, a^3 \}</math> -- the [[cyclic maximal subgroup of dihedral group:D8]]
|-
| Total (5)|| -- || -- || 8 || -- || -- 
|}

The equivalence classes up to automorphisms are:

{| class="sortable" border="1"
! Equivalence class under automorphisms in terms of <math>a,x</math> !! Geometric description of equivalence class !! Equivalence class as permutations !! Size of equivalence class !! Number of conjugacy classes in it !! Size of each conjugacy class
|-
| <math>\! \{ e \}</math> || identity element, does nothing || <math>\{ () \}</math> || 1 || 1 || 1
|-
| <math>\! \{ a^2 \}</math> || half turn || <math>\{ (1,3)(2,4) \}</math> || 1 || 1 || 1
|-
| <math>\! \{ x, ax, a^2x, a^3x \}</math> || reflections || <math>\{ (1,3)\ ,\  (2,4)\ , \ (1,4)(2,3)\ ,\  (1,2)(3,4) \}</math> || 4 || 2 || 2
|-
| <math>\! \{ a, a^3 \}</math> || rotations by odd multiples of <math>\pi/2</math> || <math>\{ (1,2,3,4)\ ,\  (1,4,3,2) \}</math> || 2 || 1 || 2
|-
| Total (4) || -- || -- || 8 || 5 || -- 
|}

<section end="conjugacy and automorphism class structure"/>

===Interpretation as dihedral group===

{{quotation|Compare with [[element structure of dihedral groups#Even degree case]]}}

Below, we consider the conjugacy class structure in terms of the interpretation ofthe group as a dihedral group of degree <math>2n</math>, where <math>n = 4</math> is even:

{| class="sortable" border="1"
! Conjugacy class type !! Size of conjugacy class (generic even <math>n</math>) !! Size of conjugacy class (<math>n = 4</math>) !! Number of conjugacy classes of this type (generic even <math>n</math>) !! Number of conjugacy classes (<math>n = 4</math>) !! Total number of elements (generic even <math>n</math>) !! Total number of elements (<math>n = 4</math>) !! Actual list of conjugacy classes
|-
| Identity element || 1 || 1 || 1 || 1 || 1 || 1 || <math>\{ e \}</math>
|-
| Non-identity element <math>a^{n/2}</math> of order two in <math>\langle a \rangle</math> || 1 || 1 || 1 || 1 || 1 || 1 || <math>\{ a^2 \}</math>
|-
| Non-identity elements in cyclic group <math>\langle a \rangle</matH>, where each element and its inverse form a conjugacy class of size two || 2 || 2 || <math>(n - 2)/2</math> || 1 || <math>n - 2</math> || 2 || <math>\{ a, a^3 \}</math>
|-
| Elements outside <math>\langle a \rangle</math>, form two conjugacy classes, one for elements of the form <math>a^{2k}x</math>, one for elements of the form <math>a^{2k+1}x</math> || <math>n/2</math> || 2 || 2 || 2 || <math>n</math> || 4 || <math>\{ x, a^2x \}, \{ ax, a^3x \}</math>
|-
! Total (--) !! -- !! -- !! <math>(n + 6)/2</math> !! 5 !! <math>2n</math> !! 8 !! --
|}

===Interpretation as unitriangular matrix group===

{{quotation|Compare with [[element structure of unitriangular matrix group of degree three over a finite field#Conjugacy class structure]]}}

We view the dihedral group of order eight as a [[unitriangular matrix group of degree three]] over [[field:F2]], which is the group under multiplication of matrices of the form:

<math>\begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix}</math>

with the entries over the field of two elements. We compare with the general theory of the conjugacy class structure of the group <math>UT(3,q)</math>, where <matH>q</matH> is the field size. We denote by <math>p</math> the prime number that is the characteristic of the field, so <matH>q</math> is a power of <math>p</math>.

Note that the letter <math>a</math> used for matrix entries has no direct relation to the letter <math>a</matH> used for group elements of <math>D_8</math>.

{| class="sortable" border="1"
! Nature of conjugacy class !! Minimal polynomial !! Size of conjugacy class (generic <math>q</math>) !! Size of conjugacy class (<math>q = 2</math>) !! Number of such conjugacy classes (generic <matH>q</math>) !! Number of such conjugacy classes (<math>q = 2</math>) !! Total number of elements (generic <math>q</math>) !! Total number of elements (<math>q = 2</math>) !! List of conjugacy classes !! Order of elements in each such conjugacy class (generic <math>q</math>) !! Order of elements in each conjugacy class (<math>q = 2</math>) !! Type of matrix
|-
| identity element || <math>t - 1</math> || 1 || 1 || 1 || 1 || 1 || 1 || <math>\{ e \}</math> || 1 || 1 || <math>a_{12} = a_{13} = a_{23}= 0</math> 
|-
| non-identity element, but central (has Jordan blocks of size one and two respectively) || <math>(t - 1)^2</math> || 1 || 1 || <math>q - 1</math> || 1 || <math>q - 1</math> || 1 || <math>\{ a^2 \}</math> || <math>p</math> || 2 || <math>a_{12} = a_{23} = 0</math> <math>a_{13} \ne 0</math>
|-
| non-central, has Jordan blocks of size one and two respectively || <math>(t - 1)^2</math> || <math>q</math> || 2 || <math>2(q - 1)</math> || 2 || <math>2q(q - 1)</math> || 4 || <math>\{ x, a^2x, \}, \{ ax, a^3x \}</math> || <math>p</math> || 2 || <math>a_{12}a_{23} = 0</math>, but not both <math>a_{12}</math> and <math>a_{23}</math> are zero 
|-
| non-central, has Jordan block of size three || <math>(t - 1)^3</math> || <math>q</math> || 2 || <math>(q - 1)^2</math> || 1 ||<math>q(q - 1)^2</math> || 2 || <math>\{ a, a^3 \}</math> || <math>p</math> if <math>p</math> odd<br>4 if <math>p = 2</math> || 4 || both <math>a_{12}</math> and <math>a_{23}</math> are nonzero
|-
! Total (--) !! -- !! -- !! -- !! <math>q^2 + q - 1</math> !! 5 !! <math>q^3</math> !! 8 !! -- !! -- !! -- !! --
|}

===Convolution algebra on conjugacy classes===

{| class="sortable" border="1"
! !! <math>\{ e \}</math> !! <math>\{ a^2 \}</math> !! <math>\{ x,a^2x \}</math> !! <math>\{ ax, a^3x \}</math> !! <math>\{ a, a^3 \}</math>
|-
| <math>\{ e \} </math> || <math>\{ e \} </math> || <math>\{ a^2 \}</math> || <math>\{ x,a^2x \}</math> || <math>\{ ax, a^3x \}</math> || <math>\{ a, a^3 \}</math>
|-
| <math>\{ a^2 \}</math> || <math>\{ a^2 \}</math> || <math>\{ e \}</math> || <math>\{ x,a^2x \}</math> || <math>\{ ax, a^3x \}</math> || <math>\{ a, a^3 \}</math>
|-
| <math>\{ x,a^2x \}</math> || <math>\{ x,a^2x \}</math> || <math>\{ x,a^2x \}</math> || <math>2 \{ e \} + 2 \{ a^2 \}</math> || <math>2 \{ a,a^3 \}</math> || <math>2 \{ ax, a^3x \}</math>
|-
| <math>\{ ax, a^3x \}</math> || <math>\{ ax, a^3x \}</math> || <math>\{ ax, a^3x \}</math> || <math>2 \{ a,a^3 \}</math> || <math>2\{ e \} + 2 \{ a^2 \}</math> || <math>2 \{ x, a^2 x \}</math>
|-
| <math>\{ a,a^3 \}</math> || <math>\{ a,a^3 \}</math> || <math>\{ a,a^3 \}</math> || <math>2\{ ax, a^3x \}</math> || <math>2 \{ x,a^2x \}</math> || <math>2 \{ e \} + 2 \{ a^2 \}</math>
|}

==Order and power information==

===Directed power graph===

Below is a trimmed version of the [[directed power graph]] of the group. There is a dark edge from one vertex to another if the latter is the square of the former. A dashed edge means that the latter is an odd power of the former. We remove all the loops.

[[File:D8directedpowergraphtrimmed.png|1000px]]

===Order statistics===

{| class="sortable" border="1"
! Number !! Elements of order exactly that number !! Number of such elements !! Number of conjugacy classes of such elements !! Number of elements whose order divides that number !! Number of conjugacy classes whose element order divides that number
|-
| 1 || <math>\! \{ e \}</math> || 1 || 1 || 1 || 1
|-
| 2 || <math>\! \{ a^2, x, ax, a^2x, a^3x \}</math> || 5 || 3 || 6 || 4
|-
| 4 || <math>\! \{ a, a^3 \}</math> || 2 || 1 || 8 || 5
|}

===Power statistics===

{| class="sortable" border="1"
! Number <math>d</math>!! <math>d^{th}</math> powers that are not <math>k^{th}</math> powers for any larger divisor <math>k</math> of the group order !! Number of such elements !! Number of conjugacy classes of such elements !! Number of  <math>d^{th}</math> powers!! Number of conjugacy classes of <math>d^{th}</math> powers
|-
| 1 || <math>\! \{ a, a^3, x, ax, a^2x, a^3x \}</math> || 6 || 3 || 8 || 5
|-
| 2 || <math>\! \{ a^2 \}</math> || 1 || 1 || 2 || 2
|-
| 4 || -- || 0 || 0 || 1 || 1
|-
| 8 || <math>\! \{ e \}</math> || 1 || 1 || 1 || 1
|}

===Order-cum-power statistics===

==Bruhat ordering==

[[File:Bruhatond8.png]]