Extended centralizer

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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.