nLab
Schur multiplier

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Group Theory

Contents

Idea

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

A group is called perfect if its abelianization is trivial, i.e., its first homology group, H 1(G;)H_1(G; \mathbb{Z}), vanishes, and a perfect group is called superperfect if its Schur multiplier also vanishes.

Let GG be a perfect discrete group. Then GG possesses a Schur cover, whose central subgroup is the Schur multiplier A uni=H 2(G;)A_{uni} = H_2(G; \mathbb{Z}).

References

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

Created on May 31, 2016 at 11:56:56. See the history of this page for a list of all contributions to it.