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 $n\geq 4$. 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 $H^{2}(A_{n};\mathbb {Z} _{2})$ 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 $H_{2}(A_{n};\mathbb {Z} )$ 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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