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.

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.