# Extended centralizer

From Groupprops

## Definition

Suppose is a group and is an element of . The **extended centralizer** of in , denoted , is the normalizer of the subset . Equivalently, it is the set of those elements of that either centralize or conjugate to .

The extended centralizer of an element is either equal to its centralizer or contains the centralizer as a subgroup of index two. The centralizer and extended centralizer are equal if and only if the element is a real element.