Schur multiplier

Definition
The Schur multiplier of a group $$G$$, denoted $$M(G)$$, is an abelian group defined in the following equivalent ways:

Equivalence of definitions

 * The equivalence of definitions (1) and (4) follows from Schur multiplier is kernel of commutator map homomorphism from exterior square to derived subgroup.
 * The equivalence of definitions (1)-(4) and (5) follows from Hopf's formula for Schur multiplier.

Facts

 * The Schur multiplier of a finite group is finite. In fact, the exponent of the Schur multiplier divides the order of the original group.
 * A Schur-trivial group is defined as a group whose Schur multiplier is trivial. It turns out that cyclic implies Schur-trivial, free implies Schur-trivial, and any finite group generated by Schur-trivial subgroups of relatively prime indices is Schur-trivial.
 * Schur multiplier of abelian group is its exterior square
 * Hopf's formula for Schur multiplier can be used to compute the Schur multiplier of a group in terms of a presentation of the group.

Group families
For various group families, the Schur multiplier can be described in terms of parameters for members of the families. The descriptions are sometimes quite complicated, so we simply provide links:

For a complete list, see Category:Group cohomology of group families.

Grouping by order
We give below the information for the group cohomology (and hence in particular, the Schur multipliers) for groups of small orders:

Related notions

 * Baer invariant is a generalization of Schur multiplier. The Schur multiplier is the Baer invariant with respect to the variety of abelian groups.
 * Nilpotent multiplier is a generalization of Schur multiplier and is a special case of the Baer invariant for the variety of groups of nilpotency class at most $$c$$.