# Every group is naturally isomorphic to its opposite group via the inverse map

From Groupprops

(Redirected from Every group is naturally isomorphic to its opposite group)

## Statement

Let be a group. Then, consider the opposite group of , which is a group with the same underlying set, and such that the binary operation is defined by:

In other words, products are taken with order reversed. Then, is isomorphic to the opposite group via the map .

This isomorphism is *natural* in the sense that it gives a natural isomorphism between the identity functor and the functor sending each group to its opposite group.

## Related facts

## Facts used

- Inverse map is involutive: This states that for all in a group, and for all in a group.