Changes

Element structure of dihedral group:D8

, 22:50, 21 June 2013
Interpretation as unitriangular matrix group
group = dihedral group:D8|
connective = of}}
{{fblike}}
We denote the identity element by $e$. The '''dihedral group''' $D_8$, sometimes called $D_4$, also called the {{dihedral group}} of order eight or the dihedral group acting on four elements, is defined by the following presentation:
$\langle x,a| a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle$
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 !! $\! e$ !! $\! a$ !! $\! a^2$ !! $\! a^3$ !! $\! x$ !! $\! ax$ !! $\! a^2x$ !! $\! a^3x$|-| $\! e$ || $\! e$ || $\! a$ || $\! a^2$ || $\! a^3$ || $\! x$ || $\! ax$ || $\! a^2x$ || $\! a^3x$|-| $\! a$ || $\! a$ ||$\! a^2$ || $\! a^3$ ||$\! e$ || $\! ax$ || $\! a^2x$ || $\! a^3x$ || $\! x$|-| $\! a^2$ || $\! a^2$ || $\! a^3$ ||$\! e$ ||$\! a$ ||$\! a^2x$ || $\! a^3x$ || $\! x$ || $\! ax$|-| $\! a^3$ || $\! a^3$ ||$\! e$ ||$\! a$ ||$\! a^2$ ||$\! a^3x$ || $\! x$ || $\! ax$ || $\! a^2x$|-| $\! x$ || $\! x$ || $\! a^3x$ || $\! a^2x$ || $\! ax$ || $\! e$ || $\! a^3$ || $\! a^2$ || $\! a$|-| $\! ax$ || $\! ax$ || $\! x$ || $\! a^3x$ || $\! a^2x$ || $\! a$|| $\! e$ || $\! a^3$ || $\! a^2$|-| $\! a^2x$ || $\! a^2x$ || $\! ax$ || $\! x$ || $\! a^3x$ || $\! a^2$ || $\! a$ || $\! e$ || $\! a^3$|-| $\! a^3x$ || $\! a^3x$ || $\! a^2x$ || $\! ax$ || $\! x$ || $\! a^3$ || $\! a^2$ || $\! a$ || $\! e$|}
<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 ($[\alpha,\beta] = \alpha^{-1}\beta^{-1}\alpha\beta$ or $[\alpha,\beta] = \alpha\beta\alpha^{-1}\beta^{-1}$) -- they are both the same map.

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

==Conjugacy class structure==
{{conjugacy class structure facts to check against}}

===General description===
<section begin="conjugacy and automorphism class structure"/>
| $\! \{ ax, a^3x \}$ || reflections about lines joining midpoints of opposite sides || $\{ (1,4)(2,3)\ , \ (1,2)(3,4) \}$ || 2 || 2 || $\{ e, a^2, ax, a^3x \}$ -- one of the [[Klein four-subgroups of dihedral group:D8]]
|-
| $\! \{ a, a^3 \}$ || rotations by odd muliples multiples of $\pi/2$ || $\{ (1,2,3,4) \ ,\ (1,4,3,2) \}$ || 2 || 4 ||$\{ e, a, a^2, a^3 \}$ -- 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 $2n$, where $n = 4$ is even:

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

$\begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix}$

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

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

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

===Convolution algebra on conjugacy classes===