# 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

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