Subgroup structure of quaternion group - Revision history

Vipul at 20:04, 29 June 2011

2011-06-29T20:04:53Z

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	==Defining functions==		==Defining functions==

	~~===Subgroup-defining functions===~~

	===Subgroup-defining functions and associated quotient-defining functions===		===Subgroup-defining functions and associated quotient-defining functions===

Vipul at 20:04, 29 June 2011

2011-06-29T20:04:09Z

@@ Line 36: / Line 36: @@
 ===Table classifying isomorphism types of subgroups===
 {| class="sortable" border="1"
-! Group name !! GAP ID !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Automorphism classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
+! Group name !! GAP ID !! [[Index of a subgroup|Index]] of subgroup !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Automorphism classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
 |-
-| [[Trivial group]] || <math>(1,1)</math> || 1 || 1 || 1 || 1 || 1
+| [[Trivial group]] || <math>(1,1)</math> || 8 || 1 || 1 || 1 || 1 || 1
 |-
-| [[Cyclic group:Z2]] || <math>(2,1)</math> ||1 || 1 || 1 || 1 || 1
+| [[Cyclic group:Z2]] || <math>(2,1)</math> || 4 || 1 || 1 || 1 || 1 || 1
 |-
-| [[Cyclic group:Z4]] || <math>(4,1)</math> || 3 || 3 || 1 || 3 || 0
+| [[Cyclic group:Z4]] || <math>(4,1)</math> || 2 || 3 || 3 || 1 || 3 || 0
 |-
-| [[Quaternion group]] || <math>(8,4)</math> || 1 || 1 || 1 || 1 || 1
+| [[Quaternion group]] || <math>(8,4)</math> || 1 || 1 || 1 || 1 || 1 || 1
 |-
 | Total || -- || 6 || 6 || 4 || 6 || 3
 |}
 ==Lattice of subgroups==

Vipul: /* Tables classifying subgroups up to automorphisms */

2011-06-29T19:54:30Z

Tables classifying subgroups up to automorphisms

← Older revision		Revision as of 19:54, 29 June 2011
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	Note that ''all'' subgroups are [[normal subgroup]]s, so the [[quotient group]] exists in all cases.		Note that ''all'' subgroups are [[normal subgroup]]s, so the [[quotient group]] exists in all cases.

			<section begin="summary"/>
	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Automorphism class of subgroups !! List of subgroups !! Isomorphism class !! Order of subgroups !! Index of subgroups !! Number of conjugacy classes (=1 iff [[automorph-conjugate subgroup]]) !! Size of each conjugacy class (=1 iff [[normal subgroup]]) !! Total number of subgroups (=1 iff [[characteristic subgroup]]) !! Isomorphism class of quotient (if exists) !! [[Nilpotency class]]		! Automorphism class of subgroups !! List of subgroups !! Isomorphism class !! Order of subgroups !! Index of subgroups !! Number of conjugacy classes (=1 iff [[automorph-conjugate subgroup]]) !! Size of each conjugacy class (=1 iff [[normal subgroup]]) !! Total number of subgroups (=1 iff [[characteristic subgroup]]) !! Isomorphism class of quotient (if exists) !! [[Nilpotency class]]
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	\| Total (4 rows) \|\| -- \|\| -- \|\| -- \|\| -- \|\| 6 \|\| -- \|\| 6 \|\| -- \|\| --		\| Total (4 rows) \|\| -- \|\| -- \|\| -- \|\| -- \|\| 6 \|\| -- \|\| 6 \|\| -- \|\| --
	\|}		\|}
			<section end="summary"/>

	===Table classifying isomorphism types of subgroups===		===Table classifying isomorphism types of subgroups===

Vipul at 15:47, 29 June 2011

2011-06-29T15:47:55Z

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	==Tables for quick information==		==Tables for quick information==

			[[File:Q8latticeofsubgroups.png\|400px]]
	{{finite p-group subgroup structure facts to check against}}		{{finite p-group subgroup structure facts to check against}}

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	\| Total \|\| -- \|\| 6 \|\| 6 \|\| 4 \|\| 6 \|\| 3		\| Total \|\| -- \|\| 6 \|\| 6 \|\| 4 \|\| 6 \|\| 3
	\|}		\|}

	~~==Lattice of subgroups==~~

	~~[[File:Q8latticeofsubgroups.png]]~~

Vipul: /* Tables classifying subgroups up to automorphisms */

2011-06-29T15:47:19Z

Tables classifying subgroups up to automorphisms

← Older revision		Revision as of 15:47, 29 June 2011
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	==Tables for quick information==		==Tables for quick information==

			{{finite p-group subgroup structure facts to check against}}

	===Tables classifying subgroups up to automorphisms===		===Tables classifying subgroups up to automorphisms===
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	\| whole group \|\| <math>\{ 1,-1,i,-i,j,-j,k,-k \}</math> \|\| [[quaternion group]] \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| [[trivial group]] \|\| 2		\| whole group \|\| <math>\{ 1,-1,i,-i,j,-j,k,-k \}</math> \|\| [[quaternion group]] \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| [[trivial group]] \|\| 2
	\|-		\|-
	\| Total (4 rows) \|\| -- \|\| -- \|\| -- \|\| -- \|\| 6 \|\| -- \|\| 6 ~~\|\| --~~ \|\| -- \|\| --		\| Total (4 rows) \|\| -- \|\| -- \|\| -- \|\| -- \|\| 6 \|\| -- \|\| 6 \|\| -- \|\| --
	\|}		\|}

Vipul at 15:46, 29 June 2011

2011-06-29T15:46:57Z

← Older revision		Revision as of 15:46, 29 June 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Automorphism class of subgroups !! List of subgroups !! Isomorphism class !! Order of subgroups !! Index of subgroups !! Number of conjugacy classes !! Size of each conjugacy class !! Total number of subgroups !! Isomorphism class of quotient (if exists) !! ~~[[Characteristic subgroup]]? !!~~ [[Nilpotency class]]		! Automorphism class of subgroups !! List of subgroups !! Isomorphism class !! Order of subgroups !! Index of subgroups !! Number of conjugacy classes (=1 iff [[automorph-conjugate subgroup]]) !! Size of each conjugacy class (=1 iff [[normal subgroup]]) !! Total number of subgroups (=1 iff [[characteristic subgroup]]) !! Isomorphism class of quotient (if exists) !! [[Nilpotency class]]
	\|-		\|-
	\| trivial subgroup \|\| <math>\{ 1 \}</math> \|\| [[trivial subgroup]] \|\| 1 \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| [[quaternion group]] ~~\|\| Yes~~ \|\| 0		\| trivial subgroup \|\| <math>\{ 1 \}</math> \|\| [[trivial subgroup]] \|\| 1 \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| [[quaternion group]] \|\| 0
	\|-		\|-
	\| [[center of quaternion group]] \|\| <math>\{ 1, -1\}</math> \|\| [[cyclic group:Z2]] \|\| 2 \|\| 4 \|\| 1 \|\| 1 \|\| 1 \|\| [[Klein four-group]] ~~\|\| Yes~~ \|\| 1		\| [[center of quaternion group]] \|\| <math>\{ 1, -1\}</math> \|\| [[cyclic group:Z2]] \|\| 2 \|\| 4 \|\| 1 \|\| 1 \|\| 1 \|\| [[Klein four-group]] \|\| 1
	\|-		\|-
	\| [[cyclic maximal subgroups of quaternion group]] \|\| <math>\{ 1,-1,i,-i \}</math><br> <math>\{ 1,-1,j,-j \}</math> <br> <math>\{ 1,-1,k,-k \}</math> \|\| [[cyclic group:Z4]] \|\| 4 \|\| 2 \|\| 3 \|\| 1 \|\| 3 \|\| [[cyclic group:Z2]] ~~\|\| No~~ \|\| 1		\| [[cyclic maximal subgroups of quaternion group]] \|\| <math>\{ 1,-1,i,-i \}</math><br> <math>\{ 1,-1,j,-j \}</math> <br> <math>\{ 1,-1,k,-k \}</math> \|\| [[cyclic group:Z4]] \|\| 4 \|\| 2 \|\| 3 \|\| 1 \|\| 3 \|\| [[cyclic group:Z2]] \|\| 1
	\|-		\|-
	\| whole group \|\| <math>\{ 1,-1,i,-i,j,-j,k,-k \}</math> \|\| [[quaternion group]] \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| [[trivial group]] ~~\|\| Yes~~ \|\| 2		\| whole group \|\| <math>\{ 1,-1,i,-i,j,-j,k,-k \}</math> \|\| [[quaternion group]] \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| [[trivial group]] \|\| 2
	\|-		\|-
	\| Total (4 rows) \|\| -- \|\| -- \|\| -- \|\| -- \|\| 6 \|\| -- \|\| 6 \|\| -- \|\| -- \|\| --		\| Total (4 rows) \|\| -- \|\| -- \|\| -- \|\| -- \|\| 6 \|\| -- \|\| 6 \|\| -- \|\| -- \|\| --

Vipul at 17:47, 22 June 2011

2011-06-22T17:47:17Z

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	group = quaternion group\|		group = quaternion group\|
	connective = of}}		connective = of}}

			The [[quaternion group]] is an 8-element group on the set:

			<math>\{ 1,-1,i,-i,j,-j,k,-k \}</math>.

			See the multiplication table of the group at [[quaternion group]] and more on the element structure at [[element structure of quaternion group]].

	==Tables for quick information==		==Tables for quick information==

Vipul at 17:46, 22 June 2011

2011-06-22T17:46:26Z

← Older revision		Revision as of 17:46, 22 June 2011
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	group = quaternion group\|		group = quaternion group\|
	connective = of}}		connective = of}}

	~~The quaternion group has four types of subgroups, when classified upto automorphism:~~

	~~# The trivial subgroup. (1)~~
	~~# The center, given by <math>\pm 1</math>. This is isomorphic to the [[subgroup::cyclic group:Z2\|cyclic group of order two]]. (1)~~
	# The three four-element cyclic subgroups, generated by the elements <math>i,j,k</math> respectively. They are all related by outer automorphisms, though no two of them are conjugate. (In fact, they're all [[normal subgroup\|normal]]). Each is isomorphic to the [[subgroup::cyclic group:Z4\|cyclic group of order four]]. (3)
	~~# The whole group. (1)~~

	==Tables for quick information==		==Tables for quick information==

Vipul: /* Tables classifying subgroups up to automorphisms */

2011-06-22T17:45:58Z

Tables classifying subgroups up to automorphisms

← Older revision		Revision as of 17:45, 22 June 2011
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	\| [[center of quaternion group]] \|\| <math>\{ 1, -1\}</math> \|\| [[cyclic group:Z2]] \|\| 2 \|\| 4 \|\| 1 \|\| 1 \|\| 1 \|\| [[Klein four-group]] \|\| Yes \|\| 1		\| [[center of quaternion group]] \|\| <math>\{ 1, -1\}</math> \|\| [[cyclic group:Z2]] \|\| 2 \|\| 4 \|\| 1 \|\| 1 \|\| 1 \|\| [[Klein four-group]] \|\| Yes \|\| 1
	\|-		\|-
	\| [[cyclic maximal subgroups of quaternion group]] \|\| <math>\{ 1,-1,i,-i \}<br> \{ 1,-1,j,-j \} <br> \{ 1,-1,k,-k \}</math> \|\| [[cyclic group:Z4]] \|\| 4 \|\| 2 \|\| 3 \|\| 1 \|\| 3 \|\| [[cyclic group:Z2]] \|\| No \|\| 1		\| [[cyclic maximal subgroups of quaternion group]] \|\| <math>\{ 1,-1,i,-i \}</math><br> <math>\{ 1,-1,j,-j \}</math> <br> <math>\{ 1,-1,k,-k \}</math> \|\| [[cyclic group:Z4]] \|\| 4 \|\| 2 \|\| 3 \|\| 1 \|\| 3 \|\| [[cyclic group:Z2]] \|\| No \|\| 1
	\|-		\|-
	\| whole group \|\| <math>\{ 1,-1,i,-i,j,-j,k,-k \}</math> \|\| [[quaternion group]] \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| [[trivial group]] \|\| Yes \|\| 2		\| whole group \|\| <math>\{ 1,-1,i,-i,j,-j,k,-k \}</math> \|\| [[quaternion group]] \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| [[trivial group]] \|\| Yes \|\| 2

Vipul: /* Tables classifying subgroups up to automorphisms */

2011-06-22T17:45:35Z

Tables classifying subgroups up to automorphisms

← Older revision		Revision as of 17:45, 22 June 2011
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	\| [[center of quaternion group]] \|\| <math>\{ 1, -1\}</math> \|\| [[cyclic group:Z2]] \|\| 2 \|\| 4 \|\| 1 \|\| 1 \|\| 1 \|\| [[Klein four-group]] \|\| Yes \|\| 1		\| [[center of quaternion group]] \|\| <math>\{ 1, -1\}</math> \|\| [[cyclic group:Z2]] \|\| 2 \|\| 4 \|\| 1 \|\| 1 \|\| 1 \|\| [[Klein four-group]] \|\| Yes \|\| 1
	\|-		\|-
	\| [[cyclic maximal subgroups of quaternion group]] \|\| <math>\{ 1,-1,i,-i \}, \{ 1,-1,j,-j \}, \{ 1,-1,k,-k \}</math> \|\| [[cyclic group:Z4]] \|\| 4 \|\| 2 \|\| 3 \|\| 1 \|\| 3 \|\| [[cyclic group:Z2]] \|\| No \|\| 1		\| [[cyclic maximal subgroups of quaternion group]] \|\| <math>\{ 1,-1,i,-i \}<br> \{ 1,-1,j,-j \} <br> \{ 1,-1,k,-k \}</math> \|\| [[cyclic group:Z4]] \|\| 4 \|\| 2 \|\| 3 \|\| 1 \|\| 3 \|\| [[cyclic group:Z2]] \|\| No \|\| 1
	\|-		\|-
	\| whole group \|\| <math>\{ 1,-1,i,-i,j,-j,k,-k \}</math> \|\| [[quaternion group]] \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| [[trivial group]] \|\| Yes \|\| 2		\| whole group \|\| <math>\{ 1,-1,i,-i,j,-j,k,-k \}</math> \|\| [[quaternion group]] \|\| 8 \|\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| [[trivial group]] \|\| Yes \|\| 2