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Element structure of dihedral group:D8

2,867 bytes added, 22:36, 16 April 2010
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group = dihedral group:D8|
connective = of}}
 
{{#lst:dihedral group:D8|multiplication table}}
 
==Conjugacy class structure==
 
{{conjugacy class structure facts to check against}}
 
{| class="sortable" border="1"
! Conjugacy class !! Size of conjugacy class !! Order of elements in conjugacy class !! Centralizer of first element of class
|-
| <math>\{ e \}</math> || 1 || 1 || whole group
|-
| <math>\{ a^2 \}</math> || 1 || 2 || whole group
|-
| <math>\{ x,a^2x \}</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> || 2 || 2 || <math>\{ e, a^2, x, a^3x \}</math> -- one of the [[Klein four-subgroups of dihedral group:D8]]
|-
| <math>\{ a, a^3 \}</math> || 2 || 4 ||<math>\{ e, a, a^2, a^3 \}</math> -- the [[cyclic maximal subgroup of dihedral group:D8]]
|}
 
The equivalence classes up to automorphisms are:
 
{| class="sortable" border="1"
! Equivalence class under automorphisms !! Size of equivalence class !! Number of conjugacy classes in it !! Size of each conjugacy class
|-
| <math>\{ e \}</math> || 1 || 1 || 1
|-
| <math>\{ a^2 \}</math> || 1 || 1 || 1
|-
| <math>\{ x, ax, a^2x, a^3x \}</math> || 4 || 2 || 2
|-
| <math>\{ a, a^3 \}</math> || 2 || 1 || 2
|}
 
===Convolution algebra on conjugacy classes===
 
{| class="wikitable" 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 statistics==
 
{| class="soritable" 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
|}
 
==Bruhat ordering==
[[File:Bruhatond8.png]]
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