Schur multiplier

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

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

No.	Shorthand	Full definition
1	second homology group	It is the second homology group for trivial group action $\! H_2(G;\mathbb{Z})$.
2	second cohomology group	This definition is valid (agreeing with other definitions) when $G$ is finite: It is the second cohomology group for trivial group action of $G$ on $\mathbb{C}^*$, i.e., the group $\! H^2(G,\mathbb{C}^*)$. Note that we could replace $\mathbb{C}^*$ by any divisible abelian group.
3	Baer invariant	It is the Baer invariant of $G$ corresponding to the subvariety of abelian groups in the variety of groups.
4	Kernel of commutator map	It is the kernel of the commutator map homomorphism from exterior square to derived subgroup, i.e., it is the kernel of the map $G \wedge G \to [G,G]$ given by $x \wedge y \mapsto [x,y]$. Here $G \wedge G$ is the exterior square of $G$ and $[G,G]$ is the derived subgroup of $G$.
5	Hopf's formula	It is given by Hopf's formula for Schur multiplier: If $G$ is isomorphic to the quotient of a free group $F$ by a normal subgroup $R$, then $M(G) = (R \cap [F,F])/[R,F]$.

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. For full proof, refer: Schur multiplier of finite group is finite
• 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.

Particular cases

Particular groups

Group	Order	Schur multiplier	Order of Schur multiplier	Explanation for Schur multiplier
trivial group	1	trivial group	1	obvious
cyclic group:Z2	2	trivial group	1	cyclic implies Schur-trivial, see also group cohomology of cyclic group:Z2 and group cohomology of finite cyclic groups
cyclic group:Z3	3	trivial group	1	cyclic implies Schur-trivial, see also group cohomology of cyclic group:Z3 and group cohomology of finite cyclic groups
cyclic group:Z4	4	trivial group	1	cyclic implies Schur-trivial, see also group cohomology of cyclic group:Z4 and group cohomology of finite cyclic groups
Klein four-group	4	cyclic group:Z2	2	see group cohomology of Klein four-group (more general information at group cohomology of elementary abelian group of prime-square order
cyclic group:Z5	5	trivial group	1	cyclic implies Schur-trivial, see also group cohomology of finite cyclic groups
symmetric group:S3	6	trivial group	1	see group cohomology of symmetric group:S3, group cohomology of symmetric groups
cyclic group:Z6	6	trivial group	1	cyclic implies Schur-trivial, see also group cohomology of finite cyclic groups
cyclic group:Z7	7	trivial group	1	cyclic implies Schur-trivial, see also group cohomology of finite cyclic groups
cyclic group:Z8	8	trivial group	1	cyclic implies Schur-trivial, see also group cohomology of finite cyclic groups
direct product of Z4 and Z2	8	cyclic group:Z2	2	see group cohomology of finite abelian groups
dihedral group:D8	8	cyclic group:Z2	2
quaternion group	8	trivial group	1
elementary abelian group:E8	8	elementary abelian group:E8	8	group cohomology of elementary abelian groups
cyclic group:Z9	9	trivial group	1	cyclic implies Schur-trivial, see also group cohomology of finite cyclic groups
elementary abelian group:E9	9	cyclic group:Z3	3	see group cohomology of elementary abelian group of prime-square order
alternating group:A4	12	cyclic group:Z2	2	see group cohomology of alternating group:A4 and group cohomology of alternating groups
symmetric group:S4	24	cyclic group:Z2	2	see group cohomology of symmetric group:S4 and group cohomology of symmetric groups
special linear group:SL(2,3)	24	trivial group	1	see group cohomology of special linear group:SL(2,3) and group cohomology of special linear group of degree two over a finite field

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:

Family	Description of Schur multiplier	Group cohomology information
cyclic group	trivial group (see cyclic implies Schur-trivial)	group cohomology of finite cyclic groups
finite abelian group	can be computed based on the invariant factors	group cohomology of finite abelian groups
symmetric group on a finite set	is trivial group for degree 3 or less, is cyclic group:Z2 for degree 4 or more	group cohomology of symmetric groups.
alternating group	is trivial group for degree 3 or less, is cyclic group:Z2 for degrees 4 or higher except degrees 6 and 7, is cyclic group:Z6 for degrees 6 and 7	group cohomology of alternating groups
special linear group of degree two over a finite field	trivial group	group cohomology of special linear group of degree two over a finite field
projective special linear group of degree two over a finite field	cyclic group:Z2 if the field has odd size, trivial group if the field has even size	group cohomology of projective special linear group of degree two over a finite field

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:

Order	Information on group cohomology
8	Group cohomology of groups of order 8#Schur multiplier and Schur covering groups
12	Group cohomology of groups of order 12#Schur multiplier and Schur covering groups
16	Group cohomology of groups of order 16#Schur multiplier and Schur covering groups
18	Group cohomology of groups of order 18#Schur multiplier and Schur covering groups
20	Group cohomology of groups of order 20#Schur multiplier and Schur covering groups
24	Group cohomology of groups of order 24#Schur multiplier and Schur covering groups
48	Group cohomology of groups of order 48#Schur multiplier and Schur covering groups

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 $c$.