Exterior square of a group

Definition

The exterior square of a group $G$ (sometimes also called the non-abelian exterior square), denoted $G \wedge G$ or $\bigwedge^2G$, can be defined in the following equivalent ways:

1. It is the exterior product of groups of $G$ with itself, where both copies of $G$ are viewed as living inside the same group $G$ as the whole group.
2. It is the derived subgroup of any Schur covering group of $G$. Note that the Schur covering groups need not be isomorphic groups, but they are isoclinic groups, so the definition is independent of the choice of Schur covering group.
3. If $G \cong F/R$ where $F$ is a free group and $R$ is a normal subgroup of $F$, it is, up to isomorphism, the same as $[F,F]/[F,R]$ (for more on this perspective, see Hopf's formula for Schur multiplier).
4. It can be defined as follows:
   • Denote by $\mathcal{F}$ the free group on the set $G \times G$.
   • For any central extension $E$ with quotient group $G$, the commutator map defines a set map $\omega:G \times G \to [E,E]$ and therefore a group homomorphism $\Omega_E: \mathcal{F} \to [E,E]$.
   • Let $\mathcal{R}$ the intersection of the kernels of all group homomorphisms $\Omega_E$ for all possible central extensions $E$. Note that although the collection of all possible homomorphisms is too big to be a set, the collection of all possible kernels is a set, so the intersection is defined.
   • The group $G \wedge G$ is defined as the quotient group $\mathcal{F}/\mathcal{R}$. The image of the freely generating element $(x,y)$ (with $x,y \in G$) is denoted $x \wedge y$.

Facts

• Commutator map is homomorphism from exterior square to derived subgroup: For any group $G$, the commutator map gives a surjective homomorphism $G \wedge G \to [G,G]$.
• Commutator map is homomorphism from exterior square to derived subgroup of central extension: Even better, if $G$ is the quotient group of a group $E$ by a central subgroup, the commutator map in $E$ descends to a homomorphism $G \wedge G \to [E,E]$.
• Schur multiplier is kernel of commutator map homomorphism from exterior square to derived subgroup: The kernel of the homomorphism $G \wedge G \to [G,G]$ mentioned above is the Schur multiplier $M(G)$, which can also be written as $H_2(G;\mathbb{Z})$. In particular, this tells us that $G \wedge G$ is a group extension with normal subgroup $M(G)$ and quotient group $[G,G]$.
• Exterior square of finite group is finite: This follows from the above and the fact that Schur multiplier of finite group is finite. The proof generalizes to showing that exterior product of finite groups is finite, which can also be used to show that tensor product of finite groups is finite.
• For an abelian group, this coincides with the exterior square of abelian group.

Particular cases

Particular groups

Group	Order	Exterior square	Order of exterior square	Schur multiplier (kernel of commutator map from exterior square) as subgroup of exterior square	Derived subgroup as a subgroup of the original group (equals the quotient of the exterior square by the Schur multiplier)	Cohomology information
trivial group	1	trivial group	1	trivial subgroup in trivial group	trivial subgroup in trivial group	--
cyclic group:Z2	2	trivial group	1	trivial subgroup in trivial group	trivial subgroup in cyclic group:Z2	group cohomology of cyclic group:Z2
cyclic group:Z3	3	trivial group	1	trivial subgroup in trivial group	trivial subgroup in cyclic group:Z3	group cohomology of cyclic group:Z3
cyclic group:Z4	4	trivial group	1	trivial subgroup in trivial group	trivial subgroup in cyclic group:Z4	group cohomology of cyclic group:Z4
Klein four-group	4	cyclic group:Z2	2	cyclic group:Z2 as a subgroup of itself	trivial subgroup in Klein four-group	group cohomology of Klein four-group
cyclic group:Z5	5	trivial group	1	trivial subgroup in trivial group	trivial subgroup in cyclic group:Z5	group cohomology of cyclic group:Z5
symmetric group:S3	6	cyclic group:Z3	3	trivial subgroup in cyclic group:Z3	A3 in S3	group cohomology of symmetric group:S3
cyclic group:Z6	6	trivial group	1	trivial subgroup in trivial group	trivial subgroup in cyclic group:Z6	group cohomology of cyclic group:Z6
cyclic group:Z7	7	trivial group	1	trivial subgroup in trivial group	trivial subgroup in cyclic group:Z7	group cohomology of cyclic group:Z7
cyclic group:Z8	8	trivial group	1	trivial subgroup in trivial group	trivial subgroup in cyclic group:Z8	group cohomology of cyclic group:Z8
direct product of Z4 and Z2	8	cyclic group:Z2	1	cyclic group:Z2 as a subgroup of itself	trivial subgroup in direct product of Z4 and Z2	group cohomology of direct product of Z4 and Z2
dihedral group:D8	8	cyclic group:Z4	4	Z2 in Z4	center of dihedral group:D8	group cohomology of dihedral group:D8
quaternion group	8	cyclic group:Z2	2	trivial subgroup in cyclic group:Z2	center of quaternion group	group cohomology of quaternion group
elementary abelian group:E8	8	elementary abelian group:E8	8	elementary abelian group:E8 as a subgroup of itself	trivial subgroup in elementary abelian group:E8	group cohomology of elementary abelian group:E8

Group families

For various families, the exterior square can be described in general terms based on the family.

Group family	Description of exterior square	Description of Schur multiplier as subgroup of exterior square	Description of derived subgroup as subgroup of the group (equals the quotient of the exterior square by the Schur multiplier)	Cohomology information
cyclic group	trivial group	trivial subgroup in trivial group	trivial subgroup in the original cyclic group	group cohomology of finite cyclic groups, group cohomology of group of integers
finite abelian group	can be computed based on invariant factors	is the whole exterior square as a subgroup of itself	trivial subgroup in the original finite abelian group	group cohomology of finite abelian groups
symmetric group on a finite set	trivial group for degree $\le 2$ cyclic group:Z3 for degree 3 double cover of alternating group for degree $\ge 4$	trivial subgroup for degree $\le 3$ center (of order two) for degree $\ge 4$	alternating group in symmetric group	group cohomology of symmetric groups
alternating group (on a finite set)	trivial group for degree $\le 3$ quaternion group for degree 4 double cover of alternating group for degree 5 and degree $\ge 8$ Schur cover of alternating group:A6 for degree 6, Schur cover of alternating group:A7 for degree 7	in all cases, the center. Has order two except for degrees: 3 (order 1), 6 (order 6), and 7 (order 6)	for degree 3, trivial subgroup for degree 4, V4 in A4 for degree $\ge 5$, the whole alternating group	group cohomology of alternating groups

Groups of a particular order

Prime numbers are not included below, because they are already covered under the group cohomology of finite cyclic groups.

Order	Information on groups	Information on exterior square and related invariants
4	groups of order 4	group cohomology of groups of order 4#Schur multiplier and Schur covering groups
6	groups of order 6	group cohomology of groups of order 6#Schur multiplier and Schur covering groups
8	groups of order 8	group cohomology of groups of order 8#Schur multiplier and Schur covering groups
16	groups of order 16	group cohomology of groups of order 16#Schur multiplier and Schur covering groups

References

• The second homology group of a group; relations among commutators by Clair Miller, Proceedings of the American Mathematical Society, Volume 3, Page 588 - 595(Year 1952): ^{Official copy (PDF)More info}: Introduced the concept without the name.
• Van Kampen theorems for diagrams of spaces by Ronald Brown and Jean-Louis Loday, Topology, Volume 26,Number 3, Page 311 - 335(Year 1987): ^{ungated copy (PDF)More info} has a discussion of the exterior square as a special case of the exterior product.
• The non-abelian tensor product of finite groups is finite by Graham Ellis, Journal of Algebra, ISSN 00218693, Volume 111, Page 203 - 205(Year 1987): ^{Official copy (Elsevier ScienceDirect, via DOI)More info} has a discussion of the exterior square as a special case of the exterior product.