nLab Schur multiplier

Idea

The Schur multiplier of a group, $G$, is $H^2(G;\mathbb{C}^{\times}) \cong H^3(G;\mathbb{Z})$. It is also the second homology group, $H_2(G; \mathbb{Z})$ with coefficients in the trivial $G$-module $\mathbb{Z}$.

A group is called perfect if its abelianization is trivial, i.e., its first homology group, $H_1(G; \mathbb{Z})$, vanishes, and a perfect group is called superperfect if its Schur multiplier also vanishes.

Let $G$ be a perfect discrete group. Then $G$ possesses a Schur cover, whose central subgroup is the Schur multiplier $A_{uni} = H_2(G; \mathbb{Z})$.

Related concepts

• group cohomology

References

• Gregory Karpilovsky 1987 The Schur Multiplier, Oxford University Press.