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.

Facts

 * If the element is not a real element, the extended centralizer equals the centralizer.
 * If the element is a real element, i.e., it is conjugate to its inverse, and it is not the identity element and not an element of order two, then the extended centralizer has the centralizer as a subgroup of index two.
 * If the element is the identity element or an involution (i.e., it has order two), then the extended centralizer equals the centralizer.