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 $$n \ge 4$$.