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

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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

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.