# Tensor square of a group

From Groupprops

## Contents

## Definition

### Arbitrary group

Suppose is a (not necessarily abelian) group. The **tensor square** of (sometimes termed the **non-abelian tensor square**) refers to the tensor product of groups where we take both the actions of on each other to be the action by conjugation.

Explicitly, it is the quotient of the free group on all formal symbols by the following relations:

- , one such relation for all
- , one such relation for all

### Abelian group

For an abelian group, the tensor square of is the same as its tensor square as a group. However, it can now also be thought of as follows: it is the tensor product of with itself in the sense of tensor product of abelian groups.

## Related notions

## Facts

- There is a natural surjective homomorphism from the tensor square of a group to the exterior square of a group (namely, send to ; the kernel is the normal closure of elements of the form ), and from that, via the commutator map, to the derived subgroup which sits inside the group.
- Kernel of natural homomorphism from tensor square to group equals third homotopy group of suspension of classifying space
- Exact sequence giving kernel of mapping from tensor square to exterior square
- Perfect implies natural mapping from tensor square to exterior square is isomorphism
- Equivalence of definitions of superperfect group: For a superperfect group, the commutator mapping defines an isomorphism from the tensor square to the whole group.

## Particular cases

### Particular groups

Group | Order | Tensor square | Kernel of homomorphism from tensor square to exterior square | Exterior square (as an abstract group) | Kernel of homomorphism from tensor square to derived subgroup | Schur multiplier (kernel of homomorphism from exterior square to derived subgroup) as a subgroup of exterior square | Derived subgroup in the whole group |
---|---|---|---|---|---|---|---|

trivial group | 1 | trivial group | trivial subgroup in trivial group | trivial group | trivial subgroup in trivial group | trivial subgroup in trivial group | trivial group |

cyclic group:Z2 | 2 | cyclic group:Z2 | cyclic group:Z2 as a subgroup of itself | trivial group | cyclic group:Z2 as a subgroup of itself | trivial subgroup of trivial group | trivial group |

cyclic group:Z3 | 3 | cyclic group:Z3 | cyclic group:Z3 as a subgroup of itself | trivial group | cyclic group:Z3 as a subgroup of itself | trivial subgroup of trivial group | trivial group |

cyclic group:Z4 | 4 | cyclic group:Z4 | cyclic group:Z4 as a subgroup of itself | trivial group | cyclic group:Z4 as a subgroup of itself | trivial subgroup of trivial group | trivial group |

Klein four-group | 4 | elementary abelian group:E16 | elementary abelian group:E8 as a subgroup of elementary abelian group:E16 | cyclic group:Z2 | elementary abelian group:E16 as a subgroup of itself | cyclic group:Z2 as a subgroup of itself | trivial group |

cyclic group:Z5 | 5 | cyclic group:Z5 | cyclic group:Z5 as a subgroup of itself | trivial group | cyclic group:Z5 as a subgroup of itself | trivial subgroup of trivial group | trivial group |

cyclic group:Z6 | 6 | cyclic group:Z6 | cyclic group:Z6 as a subgroup of itself | trivial group | cyclic group:Z6 as a subgroup of itself | trivial subgroup of trivial group | trivial group |