Schur multiplier
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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 , denoted , 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 . |
2 | second cohomology group | This definition is valid (agreeing with other definitions) when is finite: It is the second cohomology group for trivial group action of on , i.e., the group . Note that we could replace by any divisible abelian group. |
3 | Baer invariant | It is the Baer invariant of 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 given by . Here is the exterior square of and is the derived subgroup of . |
5 | Hopf's formula | It is given by Hopf's formula for Schur multiplier: If is isomorphic to the quotient of a free group by a normal subgroup , then . |
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 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 .