Double cover of alternating group - Revision history

Vipul at 03:10, 5 December 2011

2011-12-05T03:10:31Z

← Older revision		Revision as of 03:10, 5 December 2011
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			[[importance rank::3\| ]]
	==Definition==		==Definition==

Vipul: /* Particular cases */

2011-11-01T02:45:26Z

Particular cases

← Older revision		Revision as of 02:45, 1 November 2011
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	\| 6 \|\| 360 \|\| 720 \|\| [[alternating group:A6]] \|\| [[special linear group:SL(2,9)]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A6]] \|\| [[second cohomology group for trivial group action of A6 on Z2]]		\| 6 \|\| 360 \|\| 720 \|\| [[alternating group:A6]] \|\| [[special linear group:SL(2,9)]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A6]] \|\| [[second cohomology group for trivial group action of A6 on Z2]]
	\|-		\|-
	\| 7 \|\| 5040 \|\| [[alternating group:A7]] \|\| [[double cover of alternating group:A7]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A7]] \|\| [[second cohomology group for trivial group action of A7 on Z2]]		\| 7 \|\| 2520 \|\| 5040 \|\| [[alternating group:A7]] \|\| [[double cover of alternating group:A7]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A7]] \|\| [[second cohomology group for trivial group action of A7 on Z2]]
	\|-		\|-
	\| 8 \|\| 40320 \|\| [[alternating group:A8]] \|\| [[double cover of alternating group:A8]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A8]] \|\| [[second cohomology group for trivial group action of A8 on Z2]]		\| 8 \|\| 20160 \|\| 40320 \|\| [[alternating group:A8]] \|\| [[double cover of alternating group:A8]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A8]] \|\| [[second cohomology group for trivial group action of A8 on Z2]]
	\|-		\|-
	\| 9 \|\| 362880 \|\| [[alternating group:A9]] \|\| [[double cover of alternating group:A9]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A9]] \|\| [[second cohomology group for trivial group action of A9 on Z2]]		\| 9 \|\| 181440 \|\| 362880 \|\| [[alternating group:A9]] \|\| [[double cover of alternating group:A9]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A9]] \|\| [[second cohomology group for trivial group action of A9 on Z2]]
	\|}		\|}

Vipul: /* Particular cases */

2011-11-01T02:45:02Z

Particular cases

← Older revision		Revision as of 02:45, 1 November 2011
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	\| 5 \|\| 60 \|\| 120 \|\| [[alternating group:A5]] \|\| [[special linear group:SL(2,5)]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A5]] \|\| [[second cohomology group for trivial group action of A5 on Z2]]		\| 5 \|\| 60 \|\| 120 \|\| [[alternating group:A5]] \|\| [[special linear group:SL(2,5)]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A5]] \|\| [[second cohomology group for trivial group action of A5 on Z2]]
	\|-		\|-
	\| 6 \|\| 360 \|\| 720 \|\| [[special linear group:SL(2,9)]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A6]] \|\| [[second cohomology group for trivial group action of A6 on Z2]]		\| 6 \|\| 360 \|\| 720 \|\| [[alternating group:A6]] \|\| [[special linear group:SL(2,9)]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A6]] \|\| [[second cohomology group for trivial group action of A6 on Z2]]
	\|-		\|-
	\| 7 \|\| 5040 \|\| [[double cover of alternating group:A7]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A7]] \|\| [[second cohomology group for trivial group action of A7 on Z2]]		\| 7 \|\| 5040 \|\| [[alternating group:A7]] \|\| [[double cover of alternating group:A7]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A7]] \|\| [[second cohomology group for trivial group action of A7 on Z2]]
	\|-		\|-
	\| 8 \|\| 40320 \|\| [[double cover of alternating group:A8]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A8]] \|\| [[second cohomology group for trivial group action of A8 on Z2]]		\| 8 \|\| 40320 \|\| [[alternating group:A8]] \|\| [[double cover of alternating group:A8]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A8]] \|\| [[second cohomology group for trivial group action of A8 on Z2]]
	\|-		\|-
	\| 9 \|\| 362880 \|\| [[double cover of alternating group:A9]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A9]] \|\| [[second cohomology group for trivial group action of A9 on Z2]]		\| 9 \|\| 362880 \|\| [[alternating group:A9]] \|\| [[double cover of alternating group:A9]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A9]] \|\| [[second cohomology group for trivial group action of A9 on Z2]]
	\|}		\|}

Vipul at 02:44, 1 November 2011

2011-11-01T02:44:24Z

← Older revision		Revision as of 02:44, 1 November 2011
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	With the exception of the cases <math>n = 6</math> and <math>n = 7</math>, the double cover is a [[Schur covering group]]. With the further exception of <math>n = 4</math>, it is thus the [[universal central extension]].		With the exception of the cases <math>n = 6</math> and <math>n = 7</math>, the double cover is a [[Schur covering group]]. With the further exception of <math>n = 4</math>, it is thus the [[universal central extension]].

			If we consider the cohomology group <math>H^2(A_n;\mathbb{Z}_2)</math> for <math>n \ge 4</math>, the double cover corresponds to the unique element of order two in this cohomology group. For <math>n \ne 6,7</math>, this is the unique non-identity element.
	==Particular cases==		==Particular cases==

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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! <math>n</math> !! <math>n!</math> (order of the group <math>2 \cdot A_n</math>) !! The group <math>2 \cdot A_n</math> !! Is it a [[Schur covering group]] !! Is it the universal central extension?		! <math>n</math> !! <math>n!/2</math> (order of alternating group <math>A_n</math>) !! <math>n!</math> (order of the group <math>2 \cdot A_n</math>) !! Alternating group <math>A_n</math> !! The group <math>2 \cdot A_n</math> !! Is it a [[Schur covering group]] !! Is it the universal central extension? !! Cohomology information !! Cohomology group information
	\|-		\|-
	\| 4 \|\| 24 \|\| [[special linear group:SL(2,3)]] \|\| Yes \|\| No (because it's not perfect)		\| 4 \|\| 12 \|\| 24 \|\| [[alternating group:A4]] \|\| [[special linear group:SL(2,3)]] \|\| Yes \|\| No (because it's not perfect) \|\| [[group cohomology of alternating group:A4]] \|\| [[second cohomology group for trivial group action of A4 on Z2]]
	\|-		\|-
	\| 5 \|\| 120 \|\| [[special linear group:SL(2,5)]] \|\| Yes \|\| Yes		\| 5 \|\| 60 \|\| 120 \|\| [[alternating group:A5]] \|\| [[special linear group:SL(2,5)]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A5]] \|\| [[second cohomology group for trivial group action of A5 on Z2]]
	\|-		\|-
	\| 6 \|\| 720 \|\| [[special linear group:SL(2,9)]] \|\| No \|\| No		\| 6 \|\| 360 \|\| 720 \|\| [[special linear group:SL(2,9)]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A6]] \|\| [[second cohomology group for trivial group action of A6 on Z2]]
	\|-		\|-
	\| 7 \|\| 5040 \|\| [[double cover of alternating group:A7]] \|\| No \|\| No		\| 7 \|\| 5040 \|\| [[double cover of alternating group:A7]] \|\| No \|\| No \|\| [[group cohomology of alternating group:A7]] \|\| [[second cohomology group for trivial group action of A7 on Z2]]
	\|-		\|-
	\| 8 \|\| 40320 \|\| [[double cover of alternating group:A8]] \|\| Yes \|\| Yes		\| 8 \|\| 40320 \|\| [[double cover of alternating group:A8]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A8]] \|\| [[second cohomology group for trivial group action of A8 on Z2]]
	\|-		\|-
	\| 9 \|\| 362880 \|\| [[double cover of alternating group:A9]] \|\| Yes \|\| Yes		\| 9 \|\| 362880 \|\| [[double cover of alternating group:A9]] \|\| Yes \|\| Yes \|\| [[group cohomology of alternating group:A9]] \|\| [[second cohomology group for trivial group action of A9 on Z2]]
	\|}		\|}

Vipul: Created page with "==Definition== The term '''double cover of alternating group''' is used for a stem extension where the base normal subgroup is cyclic group:Z2 and the quotient group..."

2011-11-01T02:19:24Z

Created page with "==Definition== The term '''double cover of alternating group''' is used for a stem extension where the base normal subgroup is cyclic group:Z2 and the quotient group..."

New page

==Definition==

The term '''double cover of alternating group''' is used for a [[stem extension]] where the base [[normal subgroup]] is [[cyclic group:Z2]] and the quotient group is an [[alternating group]] of finite degree.

The double cover of an alternating group exists iff its degree is at least four. because the [[Schur multiplier]] <math>H_2(A_n;\mathbb{Z})</math> contains a [[cyclic group:Z2]] if and only if <math>n \ge 4</math>. Moreover, in all these cases, the double cover is unique up to isomorphism.

For <math>n \ge 5</math>, it is a [[perfect central extension]] of an [[alternating group]] of finite degree. This is because [[alternating groups on finite sets are simple]] for <math>n \ge 5</math>.

With the exception of the cases <math>n = 6</math> and <math>n = 7</math>, the double cover is a [[Schur covering group]]. With the further exception of <math>n = 4</math>, it is thus the [[universal central extension]].

==Particular cases==

The cases are for <math>n \ge 4</math>, because the [[Schur multiplier]] <math>H_2(A_n;\mathbb{Z})</math> contains a [[cyclic group:Z2]] if and only if <math>n \ge 4</math>.

{| class="sortable" border="1"
! <math>n</math> !! <math>n!</math> (order of the group <math>2 \cdot A_n</math>) !! The group <math>2 \cdot A_n</math> !! Is it a [[Schur covering group]] !! Is it the universal central extension?
|-
| 4 || 24 || [[special linear group:SL(2,3)]] || Yes || No (because it's not perfect)
|-
| 5 || 120 || [[special linear group:SL(2,5)]] || Yes || Yes
|-
| 6 || 720 || [[special linear group:SL(2,9)]] || No || No
|-
| 7 || 5040 || [[double cover of alternating group:A7]] || No || No
|-
| 8 || 40320 || [[double cover of alternating group:A8]] || Yes || Yes
|-
| 9 || 362880 || [[double cover of alternating group:A9]] || Yes || Yes
|}

==Related families==

* [[