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Element structure of dihedral group:D8

8,526 bytes added, 22:50, 21 June 2013
Interpretation as unitriangular matrix group
group = dihedral group:D8|
connective = of}}
{{fblike}}
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>
Below is the <section begin="multiplication table, with the "/> The row element is multiplied on the left and the column element is multiplied on the right.
{{#lst:dihedral group:D8|multiplication table}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]].}}
|}
<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"/>
| <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 muliples 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 || -- || --
<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===
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