Real element

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This article defines a property of elements in groups


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