Double cover of alternating group: Difference between revisions

Revision as of 02:45, 1 November 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	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
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

Related families

[[

@@ Line 21: / Line 21: @@
 | 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]]
 |}