Real element
From Groupprops
This article defines a property of elements in groups
Contents
Definition
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.
Relation with other properties
Stronger properties
Related group properties
- Ambivalent group is a group in which all elements are real elements. Symmetric groups, dihedral groups, and generalized dihedral groups are among the examples of ambivalent groups.
