If is abelian, then it is equal to its opposite group. Also, every group (not necessarily abelian) is isomorphic to its opposite group: The isomorphism (http://planetmath.org/GroupIsomorphism) is given by . More generally, any anti-automorphism gives rise to a corresponding isomorphism via , since .
Opposite groups are useful for converting a right action to a left action and vice versa. For example, if is a group that acts on on the , then a left action of on can be defined by .
|Date of creation||2013-03-22 17:09:56|
|Last modified on||2013-03-22 17:09:56|
|Last modified by||Wkbj79 (1863)|