Let be a group under the operation . The opposite group of , denoted , has the same underlying set as , and its group operation is defined by .
If is abelian, then it is equal to its opposite group. Also, every group (not necessarily abelian) is naturally isomorphic to its opposite group: An isomorphism is given by . More generally, any anti-automorphism gives rise to a corresponding isomorphism via , since
Let be an object in some category, and be a right action. Then is a left action defined by , or .