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

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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].

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