# Schur multiplier is kernel of commutator map homomorphism from exterior square to derived subgroup

## Statement

Suppose $G$ is a group. The Schur multiplier of $G$, denoted $M(G)$, which is also described as the second homology group for trivial group action $H^2(G;\mathbb{Z})$, 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:

$M(G) = \operatorname{ker}(G \wedge G \stackrel{\partial}{\to} [G,G])$

where $\partial$ sends $x \wedge y$ to $[x,y]$.

## Related facts

### Applications

• Schur muliplier of abelian group is its exterior square: This follows easily from the result of this page as follows. For an abelian group $G$, the commutator map homomorphism is trivial, so its kernel is the entire group $G \wedge G$.