# 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 contains a cyclic group:Z2 if and only if . Moreover, in all these cases, the double cover is unique up to isomorphism.

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

With the exception of the cases and , the double cover is a Schur covering group. With the further exception of , it is thus the universal central extension.

If we consider the cohomology group for , the double cover corresponds to the unique element of order two in this cohomology group. For , this is the unique non-identity element.

## Particular cases

The cases are for , because the Schur multiplier contains a cyclic group:Z2 if and only if .

## Related families

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