In
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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.
Definition
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
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
, then it is equal to its opposite group. Also, every group
(not necessarily abelian) is
naturally isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to its opposite group: An isomorphism
is given by
. More generally, any
antiautomorphism gives rise to a corresponding isomorphism
via
, since
:
Group action
Let
be an object in some category, and
be a
right action. Then
is a left action defined by
, or
.
See also
*
Opposite ring
*
Opposite category
External links
http://planetmath.org/oppositegroup
Group theory
Representation theory