Element structure of quaternion group - Revision history

Vipul at 23:29, 4 July 2011

2011-07-04T23:29:31Z

← Older revision		Revision as of 23:29, 4 July 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Conjugacy class !! Size of conjugacy class !! Order of elements in conjugacy class !! Centralizer of first element of class		! [[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 \|\| 1 \|\| whole group
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Equivalence class under automorphisms !! Size of equivalence class !! Number of conjugacy classes in it !! Size of each conjugacy class		! 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		\| <math>\! \{ 1 \}</math> \|\| 1 \|\| 1 \|\| 1 \|\| 1
	\|-		\|-
	\| <math>\! \{ -1 \}</math> \|\| 1 \|\| 1 \|\| 1		\| <math>\! \{ -1 \}</math> \|\| 1 \|\| 1 \|\| 1 \|\| 2
	\|-		\|-
	\| <math>\! \{ i,-i,j,-j,k,-k \}</math> \|\| 6 \|\| 3 \|\| 2		\| <math>\! \{ i,-i,j,-j,k,-k \}</math> \|\| 6 \|\| 3 \|\| 2 \|\| 4
	\|}		\|}
	<section end="conjugacy and automorphism class structure"/>		<section end="conjugacy and automorphism class structure"/>

Vipul at 14:49, 1 July 2011

2011-07-01T14:49:32Z

@@ Line 6: / Line 6: @@
 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.
-{{#lst:quaternion group|multiplication table}}
+<section begin="multiplication table"/>
 ==Summary==

Vipul: /* Conjugacy class structure */

2011-06-29T19:38:21Z

Conjugacy class structure

← Older revision		Revision as of 19:38, 29 June 2011
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	\|}		\|}

	==Conjugacy class structure==		==Conjugacy and automorphism class structure==

			===Conjugacy class structure===

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

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

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
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	\|}		\|}

	~~The equivalence classes up to automorphisms are:~~		===Automorphism class structure===

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
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	\| <math>\! \{ i,-i,j,-j,k,-k \}</math> \|\| 6 \|\| 3 \|\| 2		\| <math>\! \{ i,-i,j,-j,k,-k \}</math> \|\| 6 \|\| 3 \|\| 2
	\|}		\|}
			<section end="conjugacy and automorphism class structure"/>

	==Order and power information==		==Order and power information==

Vipul at 16:24, 22 May 2011

2011-05-22T16:24:14Z

← Older revision		Revision as of 16:24, 22 May 2011
Line 7:		Line 7:

	{{#lst:quaternion group\|multiplication table}}		{{#lst:quaternion group\|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 class structure==		==Conjugacy class structure==

Vipul: Created page with '{{group-specific information| group = quaternion group| information type = element structure| connective = of}} This page discusses the element structure of the [[quaternion gro...'

2010-08-09T15:12:23Z

Created page with '{{group-specific information| group = quaternion group| information type = element structure| connective = of}} This page discusses the element structure of the [[quaternion gro...'

New page

{{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.

{{#lst:quaternion group|multiplication table}}

==Conjugacy class structure==

{{conjugacy class structure facts to check against}}

{| class="sortable" border="1"
! Conjugacy class !! Size of conjugacy class !! 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>
|}

The equivalence classes up to automorphisms are:

{| class="sortable" border="1"
! Equivalence class under automorphisms !! Size of equivalence class !! Number of conjugacy classes in it !! Size of each conjugacy class
|-
| <math>\! \{ 1 \}</math> || 1 || 1 || 1
|-
| <math>\! \{ -1 \}</math> || 1 || 1 || 1
|-
| <math>\! \{ i,-i,j,-j,k,-k \}</math> || 6 || 3 || 2
|}

==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
|}