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Element structure of dihedral group:D8

109 bytes added, 22:50, 21 June 2013
Interpretation as unitriangular matrix group
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 a_{12} & b a_{13} \\ 0 & 1 & c 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>.
We compare with
{| class="sortable" border="1"
! Nature of conjugacy class !! Type of matrix !! 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>a = b = c = 0</math> || <math>x 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>a = c = 0</math> <math>b \ne 0</math> || <math>(x 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>ac = 0</math>, but not both <math>a</math> and <math>c</math> are zero || <math>(x t - 1)^2</math> || <math>q</math> || 2 || <math>2q 2(q - 21)</math> || 2 || <math>2q(q(2q - 21)</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 || both <math>a</math> and <math>c</math> are nonzero || <math>(x 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!! -- !! -- !! -- !! --
|}
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