Open main menu

Groupprops β

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:

  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

Related group properties