# Schur multiplier

## Contents

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 $M(G) = (R \cap [F,F])/[R,F]$.

## 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:

## 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 $c$.