Real element

This article defines a property of elements in groups

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.

Relation with other properties

Stronger properties

• Involution
• Strongly real element
• 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.