Double cover of alternating group: Difference between revisions

Latest revision as of 03:10, 5 December 2011

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 $H_{2} (A_{n}; Z)$ contains a cyclic group:Z2 if and only if $n \geq 4$ . Moreover, in all these cases, the double cover is unique up to isomorphism.

For $n \geq 5$ , it is a perfect central extension of an alternating group of finite degree. This is because alternating groups on finite sets are simple for $n \geq 5$ .

With the exception of the cases $n = 6$ and $n = 7$ , the double cover is a Schur covering group. With the further exception of $n = 4$ , it is thus the universal central extension.

If we consider the cohomology group $H^{2} (A_{n}; Z_{2})$ for $n \geq 4$ , the double cover corresponds to the unique element of order two in this cohomology group. For $n \neq 6, 7$ , this is the unique non-identity element.

Particular cases

The cases are for $n \geq 4$ , because the Schur multiplier $H_{2} (A_{n}; Z)$ contains a cyclic group:Z2 if and only if $n \geq 4$ .

$n$	$n! / 2$ (order of alternating group $A_{n}$ )	$n!$ (order of the group $2 \cdot A_{n}$ )	Alternating group $A_{n}$	The group $2 \cdot A_{n}$	Is it a Schur covering group	Is it the universal central extension?	Cohomology information	Cohomology group information
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	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	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	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	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	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

Related families

[[

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+[[importance rank::3| ]]
 ==Definition==
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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]].
+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==
@@ Line 14: / Line 16: @@
 {| 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 || [[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
+| 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 || [[double cover of alternating group:A8]] || Yes || Yes
+| 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 || [[double cover of alternating group:A9]] || Yes || Yes
+| 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]]
 |}