# Difference between revisions of "Element structure of dihedral group:D8"

View element structure of particular groups | View other specific information about dihedral group:D8

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:

$\langle x,a| a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle$

The row element is multiplied on the left and the column element is multiplied on the right.

Element $\! e$ $\! a$ $\! a^2$ $\! a^3$ $\! x$ $\! ax$ $\! a^2x$ $\! a^3x$
$\! e$ $\! e$ $\! a$ $\! a^2$ $\! a^3$ $\! x$ $\! ax$ $\! a^2x$ $\! a^3x$
$\! a$ $\! a$ $\! a^2$ $\! a^3$ $\! e$ $\! ax$ $\! a^2x$ $\! a^3x$ $\! x$
$\! a^2$ $\! a^2$ $\! a^3$ $\! e$ $\! a$ $\! a^2x$ $\! a^3x$ $\! x$ $\! ax$
$\! a^3$ $\! a^3$ $\! e$ $\! a$ $\! a^2$ $\! a^3x$ $\! x$ $\! ax$ $\! a^2x$
$\! x$ $\! x$ $\! a^3x$ $\! a^2x$ $\! ax$ $\! e$ $\! a^3$ $\! a^2$ $\! a$
$\! ax$ $\! ax$ $\! x$ $\! a^3x$ $\! a^2x$ $\! a$ $\! e$ $\! a^3$ $\! a^2$
$\! a^2x$ $\! a^2x$ $\! ax$ $\! x$ $\! a^3x$ $\! a^2$ $\! a$ $\! e$ $\! a^3$
$\! a^3x$ $\! a^3x$ $\! a^2x$ $\! ax$ $\! x$ $\! a^3$ $\! a^2$ $\! a$ $\! e$

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.

## Summary

Item Value
order of the whole group (total number of elements) 8
conjugacy class sizes 1,1,2,2,2
maximum: 2, number of conjugacy classes: 5, lcm: 2
order statistics 1 of order 1, 5 of order 2, 2 of order 4
maximum: 4, lcm (exponent of the whole group): 4

## Elements

Below, we list all the elements, also giving the interpretation of each element under the geometric description of the dihedral group as the symmetries of a 4-gon, and for the corresponding permutation representation (see D8 in S4). Note that for different conventions, one can obtain somewhat different correspondences, so this may not match up with other correspondences elsewhere. Note that the descriptions below assume the left action convention for functions and the corresponding convention for composition, and hence some of the entries may become different if you adopt the right action convention.:

Element in terms of $a$ and $x$ Geometric description Permutation on vertices Order of the element
$e$ (identity element) does nothing, i.e., leaves the square invariant $()$ 1
$a$ rotation by angle of $\pi/2$ (i.e., $90\,^\circ$) counterclockwise $(1,2,3,4)$ 4
$a^2$ rotation by angle of $\pi$ (i.e., $180\,^\circ$), also called a half turn $(1,3)(2,4)$ 2
$a^3$ rotation by angle of $3\pi/2$ (i.e., $270\,^\circ$) counter-clockwise, or equivalently, by $\pi/2$ (i.e., $90\,^\circ$) clockwise $(1,4,3,2)$ 4
$x$ reflection about the diagonal joining vertices "2" and "4" $(1,3)$ 2
$ax = xa^3$ reflection about the line joining midpoints of opposite sides "14" and "23" $(1,4)(2,3)$ 2
$a^2x$ reflection about the diagonal joining vertices "1" and "3" $(2,4)$ 2
$a^3x = xa$ reflection about the line joining midpoints of opposite sides "12" and "34" $(1,2)(3,4)$ 2

## 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.

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

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

### General description

Conjugacy class in terms of $a,x$ 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
$\! \{ e \}$ identity element, does nothing $\{ () \}$ 1 1 whole group
$\! \{ a^2 \}$ half turn, rotation by $\pi$ $\{ (1,3)(2,4) \}$ 1 2 whole group
$\! \{ x,a^2x \}$ reflections about diagonals $\{ (1,3), (2,4) \}$ 2 2 $\{ e, a^2, x, a^2x \}$ -- one of the Klein four-subgroups of dihedral group:D8
$\! \{ ax, a^3x \}$ 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 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 -- --

The equivalence classes up to automorphisms are:

Equivalence class under automorphisms in terms of $a,x$ Geometric description of equivalence class Equivalence class as permutations Size of equivalence class Number of conjugacy classes in it Size of each conjugacy class
$\! \{ e \}$ identity element, does nothing $\{ () \}$ 1 1 1
$\! \{ a^2 \}$ half turn $\{ (1,3)(2,4) \}$ 1 1 1
$\! \{ x, ax, a^2x, a^3x \}$ reflections $\{ (1,3)\ ,\ (2,4)\ , \ (1,4)(2,3)\ ,\ (1,2)(3,4) \}$ 4 2 2
$\! \{ a, a^3 \}$ rotations by odd multiples of $\pi/2$ $\{ (1,2,3,4)\ ,\ (1,4,3,2) \}$ 2 1 2
Total (4) -- -- 8 5 --

### Interpretation as dihedral group

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:

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 $\langle a \rangle$, where each element and its inverse form a conjugacy class of size two 2 2 $(n - 2)/2$ 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

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 $q$ is the field size. We denote by $p$ the prime number that is the characteristic of the field, so $q$ is a power of $p$.

Note that the letter $a$ used for matrix entries has no direct relation to the letter $a$ used for group elements of $D_8$.

Nature of conjugacy class Minimal polynomial Size of conjugacy class (generic $q$) Size of conjugacy class ($q = 2$) Number of such conjugacy classes (generic $q$) 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
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

$\{ e \}$ $\{ a^2 \}$ $\{ x,a^2x \}$ $\{ ax, a^3x \}$ $\{ a, a^3 \}$
$\{ e \}$ $\{ e \}$ $\{ a^2 \}$ $\{ x,a^2x \}$ $\{ ax, a^3x \}$ $\{ a, a^3 \}$
$\{ a^2 \}$ $\{ a^2 \}$ $\{ e \}$ $\{ x,a^2x \}$ $\{ ax, a^3x \}$ $\{ a, a^3 \}$
$\{ x,a^2x \}$ $\{ x,a^2x \}$ $\{ x,a^2x \}$ $2 \{ e \} + 2 \{ a^2 \}$ $2 \{ a,a^3 \}$ $2 \{ ax, a^3x \}$
$\{ ax, a^3x \}$ $\{ ax, a^3x \}$ $\{ ax, a^3x \}$ $2 \{ a,a^3 \}$ $2\{ e \} + 2 \{ a^2 \}$ $2 \{ x, a^2 x \}$
$\{ a,a^3 \}$ $\{ a,a^3 \}$ $\{ a,a^3 \}$ $2\{ ax, a^3x \}$ $2 \{ x,a^2x \}$ $2 \{ e \} + 2 \{ a^2 \}$

## Order and power information

### Directed power graph

Below is a trimmed version of the directed power graph of the group. There is a dark edge from one vertex to another if the latter is the square of the former. A dashed edge means that the latter is an odd power of the former. We remove all the loops.

### Order statistics

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 $\! \{ e \}$ 1 1 1 1
2 $\! \{ a^2, x, ax, a^2x, a^3x \}$ 5 3 6 4
4 $\! \{ a, a^3 \}$ 2 1 8 5

### Power statistics

Number $d$ $d^{th}$ powers that are not $k^{th}$ powers for any larger divisor $k$ of the group order Number of such elements Number of conjugacy classes of such elements Number of $d^{th}$ powers Number of conjugacy classes of $d^{th}$ powers
1 $\! \{ a, a^3, x, ax, a^2x, a^3x \}$ 6 3 8 5
2 $\! \{ a^2 \}$ 1 1 2 2
4 -- 0 0 1 1
8 $\! \{ e \}$ 1 1 1 1