# Real element

*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.