Double cover of alternating group
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.
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.