Real element

In group theory, a discipline within modern algebra, an element $$x$$ of a group $$G$$ is called a real element of $$G$$ if it belongs to the same conjugacy class as its inverse $$x^{-1}$$, that is, if there is a $$g$$ in $$G$$ with $$x^g = x^{-1}$$, where $$x^g$$ is defined as $$g^{-1} \cdot x \cdot g$$. An element $$x$$ of a group $$G$$ is called strongly real if there is an involution $$t$$ with $$x^t = x^{-1}$$.

An element $$x$$ of a group $$G$$ is real if and only if for all representations $$\rho$$ of $$G$$, the trace $$\mathrm{Tr}(\rho(g))$$ of the corresponding matrix is a real number. In other words, an element $$x$$ of a group $$G$$ is real if and only if $$\chi(x)$$ is a real number for all characters $$\chi$$ of $$G$$.

A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group $$S_n$$ of any degree $$n$$ is ambivalent.

Properties
A group with real elements other than the identity element necessarily is of even order.

For a real element $$x$$ of a group $$G$$, the number of group elements $$g$$ with $$x^g = x^{-1}$$ is equal to $$\left|C_G(x)\right|$$, where $$C_G(x)$$ is the centralizer of $$x$$,


 * $$\mathrm{C}_G(x) = \{ g \in G\mid x^g = x \}$$.

Every involution is strongly real. Furthermore, every element that is the product of two involutions is strongly real. Conversely, every strongly real element is the product of two involutions.

If $ x \ne e$ and $$x$$ is real in $$G$$ and $$\left|C_G(x)\right|$$ is odd, then $$x$$ is strongly real in $$G$$.

Extended centralizer
The extended centralizer of an element $$x$$ of a group $$G$$ is defined as


 * $$\mathrm{C}^*_G(x) = \{ g \in G\mid x^g = x \lor x^g = x^{-1} \},$$

making the extended centralizer of an element $$x$$ equal to the normalizer of the set $\left\{x, x^{-1}\right\}$.

The extended centralizer of an element of a group $$G$$ is always a subgroup of $$G$$. For involutions or non-real elements, centralizer and extended centralizer are equal. For a real element $$x$$ of a group $$G$$ that is not an involution,


 * $$\left|\mathrm{C}^*_G(x):\mathrm{C}_G(x)\right| = 2.$$