Tensor square of a group

Definition

Arbitrary group

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

Explicitly, it is the quotient of the free group on all formal symbols $\{g\otimes h\mid g,h\in G\}$ by the following relations:

• $(g_{1}g_{2})\otimes h=(g_{1}g_{2}g_{1}^{-1}\otimes g_{1}hg_{1}^{-1})(g_{1}\otimes h)$, one such relation for all $g_{1},g_{2},h\in G$
• $g\otimes (h_{1}h_{2})=(g\otimes h_{1})(h_{1}gh_{1}^{-1}\otimes h_{1}h_{2}h_{1}^{-1})$, one such relation for all $g,h_{1},h_{2}\in G$

Abelian group

For $G$ an abelian group, the tensor square of $G$ 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 $G$ with itself in the sense of tensor product of abelian groups.

Related notions

• Exterior square of a group
• Tensor power of a group

Facts

• There is a natural surjective homomorphism from the tensor square of a group to the exterior square of a group (namely, send $x\otimes y$ to $x\wedge y$; the kernel is the normal closure of elements of the form $x\otimes x$), 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