Extended centralizer

Definition

Suppose $G$ is a group and $x$ is an element of $G$. The extended centralizer of $x$ in $G$, denoted $C_G^{*}(x)$, is the normalizer of the subset $\{x,x^{-1}\}$. Equivalently, it is the set of those elements of $G$ that either centralize $x$ or conjugate $x$ to $x^{-1}$.

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.