This article defines a property of elements in groups
An element in a group is said to be real if it satisfies the following equivalent conditions:
- It is conjugate to its inverse.
- Its extended centralizer in the whole group equals its centralizer in the whole group.
- (For finite groups): For any representation of the group over the complex numbers, the character has a real value at that element.
A group in which all elements are real is termed an ambivalent group.