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Element structure of dihedral group:D8

508 bytes added, 02:21, 22 June 2011
Conjugacy class structure
{{conjugacy class structure facts to check against}}
<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
{| 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
|}
<section end="conjugacy and automorphism class structure"/>
===Convolution algebra on conjugacy classes===
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