Schur multiplier is kernel of commutator map homomorphism from exterior square to derived subgroup
Suppose is a group. The Schur multiplier of , denoted , which is also described as the second homology group for trivial group action , can be computed as follows: it is the kernel of the homomorphism from the exterior square to the derived subgroup given by the commutator map. Explicitly:
where sends to .
- Perfect implies natural mapping from tensor square to exterior square is isomorphism
- Exact sequence giving kernel of mapping from tensor square to exterior square
- Kernel of natural homomorphism from tensor square to group equals third homotopy group of suspension of classifying space
- Schur muliplier of abelian group is its exterior square: This follows easily from the result of this page as follows. For an abelian group , the commutator map homomorphism is trivial, so its kernel is the entire group .