Real element

Definition
An element in a group is said to be real if it satisfies the following equivalent conditions:


 * 1) It is conjugate to its inverse.
 * 2) Its extended centralizer in the whole group equals its centralizer in the whole group.
 * 3) (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.

Stronger properties

 * Weaker than::Involution
 * Weaker than::Strongly real element
 * Weaker than::Rational element

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.