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