Variant of Hopf's formula for Schur multiplier for nilpotent group that uses the free nilpotent group of class one more

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Statement

Suppose G is a nilpotent group of nilpotency class c. We can calculate the Schur multiplier of G as follows. Let F be a free nilpotent group of class c + 1 and R a normal subgroup of F such that G \cong F/R. The Schur multiplier M(G) can be computed as:

M(G) \cong (R \cap [F,F])/[F,R]

This is a variant of Hopf's formula for Schur multiplier. The original version of the formula stipulates that F must be a free group. Note that, when generalizing, what's crucial is to have a class of one more than the class of G, because we need a quotient that is big enough to be sensitive to [F,R].

Corresponding formula for exterior square

Suppose G is a nilpotent group of nilpotency class c. We can calculate the Schur multiplier of G as follows. Let F be a free nilpotent group of class c + 1 and R a normal subgroup of F such that G \cong F/R. The exterior square G \wedge G can be computed as:

G \wedge G \cong [F,F]/[F,R]