Double cover of alternating group

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 $H_2(A_n;\mathbb{Z})$ contains a cyclic group:Z2 if and only if $n \ge 4$ . Moreover, in all these cases, the double cover is unique up to isomorphism.

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

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 $H^2(A_n;\mathbb{Z}_2)$ for $n \ge 4$ , the double cover corresponds to the unique element of order two in this cohomology group. For $n \ne 6,7$ , this is the unique non-identity element.

Particular cases

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

$n$	$n!/2$ (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