View source for Element structure of quaternion group

{{group-specific information|
group = quaternion group|
information type = element structure|
connective = of}}

This page discusses the element structure of the [[quaternion group]]. Notation for the quaternion group differs somewhat from notation for most groups. The multiplication table that we use throughout to identify elements is given below.

<section begin="multiplication table"/>

In the table below, the row element is multiplied on the left and the column element on the right.
{| class="sortable" border="1"
!Element !! <math>\! 1</math> !! <math>\! -1</math> !! <math>\! i</math> !! <math>\! -i</math> !! <math>\! j</math> !! <math>\! -j</math> !! <math>\! k</math> !! <math>\! -k</math>
|-
| <math>\! 1</math> || <math>\! 1</math> || <math>\! -1</math> || <math>\! i</math> || <math>\! -i</math> || <math>\! j</math> || <math>\! -j</math> || <math>\! k</math> || <math>\! -k</math>
|-
| <math>\! -1</math> || <math>\! -1</math> || <math>\! 1</math> || <math>\! -i</math> || <math>\! i</math> || <math>\! -j</math> || <math>\! j</math> || <math>\! -k</math> || <math>\! k</math>
|-
| <math>\! i</math> || <math>\! i</math> || <math>\! -i</math> || <math>\! -1</math> || <math>\! 1</math> || <math>\! k</math> || <math>\! -k</math> || <math>\! -j</math> || <math>\! j</math>
|-
| <math>\! -i</math> || <math>\! -i</math> || <math>\! i</math> || <math>\! 1</math> || <math>\! -1</math> || <math>\! -k</math> || <math>\! k</math> || <math>\! j</math> || <math>\! -j</math>
|-
| <math>\! j</math> || <math>\! j</math> || <math>\! -j</math> || <math>\! -k</math> || <math>\! k</math> || <math>\! -1</math> || <math>\! 1</math> || <math>\! i</math> || <math>\! -i</math>
|-
| <math>\! -j</math> || <math>\! -j</math> || <math>\! j</math> || <math>\! k</math> || <math>\! -k</math> || <math>\! 1</math> || <math>\! -1</math> || <math>\! -i</math> || <math>\! i</math>
|-
| <math>\! k</math> || <math>\! k</math> || <math>\! -k</math> || <math>\! j</math> || <math>\! -j</math> || <math>\! -i</math> || <math>\! i</math> || <math>\! -1</math> || <math>\! 1</math>
|-
| <math>\! -k</math> || <math>\! -k</math> || <math>\! k</math> || <math>\! -j</math> || <math>\! j</math> || <math>\! i</math> || <math>\! -i</math> || <math>\! 1</math> || <math>\! -1</math>
|}

<section end="multiplication table"/>
==Summary==

{| class="sortable" border="1"
! Item !! Value
|-
| [[order of a group|order]] of the whole group (total number of elements) || 8
|-
| [[conjugacy class size statistics of a finite group|conjugacy class sizes]] || 1,1,2,2,2<br>[[maximum conjugacy class size|maximum]]: 2, [[number of conjugacy classes]]: 5, [[lcm of conjugacy class sizes|lcm]]: 2
|-
| [[order statistics of a finite group|order statistics]] || 1 of order 1, 1 of order 2, 6 of order 4<br>[[maximum of element orders|maximum]]: 4, [[exponent of a group|lcm (exponent of the whole group)]]: 4
|}

==Conjugacy and automorphism class structure==

===Conjugacy class structure===

{{conjugacy class structure facts to check against}}

<section begin="conjugacy and automorphism class structure"/>

{| class="sortable" border="1"
! [[Conjugacy class]] !! [[Conjugacy class size statistics of a finite group|Size of conjugacy class]] !! [[Order of an element|Order of elements]] in conjugacy class !! Centralizer of first element of class
|-
| <math>\! \{ 1 \}</math> || 1 || 1 || whole group
|-
| <math>\! \{ -1 \}</math> || 1 || 2 || whole group
|-
| <math>\! \{ i,-i \}</math> || 2 || 4 || <math>\{ 1,-1,i,-i \}</math>, same as <math>\langle i \rangle</math>
|-
| <math>\! \{ j,-j \}</math> || 2 || 4 || <math>\{ 1,-1,j,-j\}</math> -- same as <math>\langle j \rangle</math>
|-
| <math>\! \{ k,-k \}</math> || 2 || 4 ||<math>\{ 1,-1,k,-k \}</math> -- same as <math>\langle k \rangle</math>
|}

===Automorphism class structure===

{| class="sortable" border="1"
! Equivalence class (orbit) under action of automorphisms !! Size of equivalence class (orbit) !! Number of conjugacy classes in it !! Size of each conjugacy class !! Order of elements
|-
| <math>\! \{ 1 \}</math> || 1 || 1 || 1 || 1
|-
| <math>\! \{ -1 \}</math> || 1 || 1 || 1 || 2
|-
| <math>\! \{ i,-i,j,-j,k,-k \}</math> || 6 || 3 || 2 || 4
|}
<section end="conjugacy and automorphism class structure"/>

==Order and power information==

===Directed power graph===

Below is a trimmed version of the [[directed power graph]] of the group. There is an arrow from one vertex to another if the latter is the square of the former. We do not draw a loop at the identity element.

[[File:Q8directedpowergraphtrimmed.png|1000px]]

===Order statistics===

{| class="sortable" border="1"
! 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 || <math>\! \{ 1 \}</math> || 1 || 1 || 1 || 1
|-
| 2 || <math>\! \{ -1 \}</math> || 1 || 1 || 2 || 2
|-
| 4 || <math>\| \{ i,-i,j,-j,k,-k \}</math> || 6 || 3 || 8 || 5
|}

===Power statistics===

{| class="sortable" border="1"
! Number <math>d</math>!! <math>d^{th}</math> powers that are not <math>k^{th}</math> powers for any larger divisor <math>k</math> of the group order !! Number of such elements !! Number of conjugacy classes of such elements !! Number of  <math>d^{th}</math> powers!! Number of conjugacy classes of <math>d^{th}</math> powers
|-
| 1 || <math>\! \{ i,-i,j,-j,k,-k \}</math> || 6 || 3 || 8 || 5
|-
| 2 || <math>\! \{ -1 \}</math> || 1 || 1 || 2 || 2
|-
| 4 || -- || 0 || 0 || 1 || 1
|-
| 8 || <math>\! \{ 1 \}</math> || 1 || 1 || 1 || 1
|}