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

Statement

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

$x*y:=yx$

In other words, products are taken with order reversed. Then, $G$ is isomorphic to the opposite group via the map $g\mapsto g^{-1}$.

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

• Left and right coset spaces are naturally isomorphic

Facts used

1. Inverse map is involutive: This states that $(xy{)}^{-1}=y^{-1}{x}^{-1}$ for all $x,y$ in a group, and $(x^{-1}{)}^{-1}=x$ for all $x$ in a group.