Opposite group

In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action.

Monoids, groups, rings, and algebras can be viewed as categories with a single object.

The construction of the opposite category generalizes the opposite group, opposite ring, etc.

be a group under the operation

The opposite group of

, denoted

, has the same underlying set as

, and its group operation

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

φ :

More generally, any antiautomorphism

gives rise to a corresponding isomorphism

be an object in some category, and

{\displaystyle \rho :G\to \mathrm {Aut} (X)}

{\displaystyle \rho ^{\mathrm {op} }:G^{\mathrm {op} }\to \mathrm {Aut} (X)}

is a left action defined by