Exterior square of a group

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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

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