# Schur multiplier of abelian group is its exterior square

## Statement

Suppose $G$ is an abelian group. The Schur multiplier of $G$, denoted $M(G)$, which is the same as the second homology group for trivial group action $H_2(G;\mathbb{Z})$ is isomorphic to the group $\bigwedge^2 G$, defined as the exterior square of $G$ viewed as a $\mathbb{Z}$-module.

## Related facts

### Generalizations

Generalization is to ... Statement How this is a special case
arbitrary group instead of abelian group Schur multiplier is kernel of commutator map homomorphism from exterior square to derived subgroup when the group is an abelian group, the commutator map is the trivial map and hence the kernel of the homomorphism is the whole exterior square.
nilpotent multiplier instead of Schur multiplier nilpotent multiplier of abelian group is graded component of free Lie ring The Schur multiplier is the $c$-nilpotent multiplier for $c = 1$. Thus, it equals the $(c+1)^{th}$ graded component of the free Lie ring. In Lie theory, the relations in degree two are generated by the alternating condition, so the graded component is precisely the exterior square.