# Tensor product of groups is commutative up to natural isomorphism

## Contents

## Statement

Suppose and are (not necessarily abelian) groups with a compatible pair of actions and . We can define the tensor product of groups and also the tensor product of groups . The claim is that these tensor products are isomorphic groups with a natural isomorphism given as follows on a generating set:

Note that this natural isomorphism is self-inverse, i.e., the natural isomorphism from to is the inverse map to the similarly defined natural isomorphism from to .

## Related facts

## Proof

### Proof idea

Consider the two axioms that define the tensor product:

Note that apart from interchanging the roles of and , the other key difference between the two axioms is that the order of multiplication on the respective right sides differs. Since the inverse map is involutive, this explains why works.