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