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

From Groupprops
Jump to: navigation, search
(Conjugacy class structure)
Line 64: Line 64:
 
| <math>\! \{ ax, a^3x \}</math> || reflections about lines joining midpoints of opposite sides || <math>\{ (1,4)(2,3)\ , \ (1,2)(3,4) \}</math> || 2 || 2 || <math>\{ e, a^2, ax, a^3x \}</math> -- one of the [[Klein four-subgroups of dihedral group:D8]]
 
| <math>\! \{ ax, a^3x \}</math> || reflections about lines joining midpoints of opposite sides || <math>\{ (1,4)(2,3)\ , \ (1,2)(3,4) \}</math> || 2 || 2 || <math>\{ e, a^2, ax, a^3x \}</math> -- one of the [[Klein four-subgroups of dihedral group:D8]]
 
|-
 
|-
| <math>\! \{ a, a^3 \}</math> || rotations by odd muliples of <math>\pi/2</math> || <math>\{ (1,2,3,4), (1,4,3,2) \}</math> || 2 || 4 ||<math>\{ e, a, a^2, a^3 \}</math> -- the [[cyclic maximal subgroup of dihedral group:D8]]
+
| <math>\! \{ a, a^3 \}</math> || rotations by odd muliples of <math>\pi/2</math> || <math>\{ (1,2,3,4) \ ,(1,4,3,2) \}</math> || 2 || 4 ||<math>\{ e, a, a^2, a^3 \}</math> -- the [[cyclic maximal subgroup of dihedral group:D8]]
 
|-
 
|-
 
| Total (5)|| -- || -- || 8 || -- || --  
 
| Total (5)|| -- || -- || 8 || -- || --  
Line 78: Line 78:
 
| <math>\! \{ a^2 \}</math> || half turn || <math>\{ (1,3)(2,4) \}</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>\! \{ 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
+
| <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
 
|-
 
|-
 
| Total (4) || -- || -- || 8 || 5 || --  
 
| Total (4) || -- || -- || 8 || 5 || --  

Revision as of 02:27, 22 June 2011

This article gives specific information, namely, element structure, about a particular group, namely: 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

Below is the multiplication table, with the row element multiplied on the left and the column element on the right.


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

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


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 muliples 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 --


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.

D8directedpowergraphtrimmed.png

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

Order-cum-power statistics

Bruhat ordering

Bruhatond8.png