Schur multiplier

The Schur multiplier of a group is a single abelian group that packages all the ways the group can be a central extension. Schur introduced it to answer a representation-theoretic puzzle: a finite group can sometimes be represented only projectively, where matrices multiply up to a scalar, and the multiplier measures exactly the obstruction to straightening these projective representations into ordinary ones. It is the canonical invariant attached to a group's hidden “twist budget”.

Concretely the Schur multiplier of a group G is the second homology group M(G) = H_2(G, Z) with integer coefficients and trivial action. For a finite group it is finite, and by the universal coefficient theorem it is isomorphic to H^2(G, C^*), the group classifying central extensions of G by the circle, equivalently the classes of projective representations. The multiplier is functorial and behaves well under products: M(G × H) contains M(G) ⊕ M(H) ⊕ (G^ab ⊗ H^ab).

For a perfect group G (one equal to its own commutator subgroup) there is a universal central extension 1 -> M(G) -> Ĝ -> G -> 1, and Ĝ is the unique central extension that is itself perfect and covers all others; M(G) is its kernel, the fundamental group of G in the algebraic sense. The Schur multiplier appears throughout finite group theory — it is part of the data in the classification of finite simple groups, where each simple group carries a known multiplier controlling its covers.