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.
Let
G
be a group under the operation
∗
. The opposite group of
G
, denoted
G
o
p
, has the same underlying set as
G
, and its group operation
∗
′
is defined by
g
1
∗
′
g
2
=
g
2
∗
g
1
.
If
G
is abelian, then it is equal to its opposite group. Also, every group
G
(not necessarily abelian) is naturally isomorphic to its opposite group: An isomorphism
φ
:
G
→
G
o
p
is given by
φ
(
x
)
=
x
−
1
. More generally, any antiautomorphism
ψ
:
G
→
G
gives rise to a corresponding isomorphism
ψ
′
:
G
→
G
o
p
via
ψ
′
(
g
)
=
ψ
(
g
)
, since
ψ
′
(
g
∗
h
)
=
ψ
(
g
∗
h
)
=
ψ
(
h
)
∗
ψ
(
g
)
=
ψ
(
g
)
∗
′
ψ
(
h
)
=
ψ
′
(
g
)
∗
′
ψ
′
(
h
)
.
Let
X
be an object in some category, and
ρ
:
G
→
A
u
t
(
X
)
be a right action. Then
ρ
o
p
:
G
o
p
→
A
u
t
(
X
)
is a left action defined by
ρ
o
p
(
g
)
x
=
ρ
(
g
)
x
, or
g
o
p
x
=
x
g
.
