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

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

A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group ${\displaystyle S_{n}}$ of any degree ${\displaystyle n}$ is ambivalent.

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

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

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 ${\displaystyle x\neq e}$ and ${\displaystyle x}$ is real in ${\displaystyle G}$ and ${\displaystyle \left|C_{G}(x)\right|}$ is odd, then ${\displaystyle x}$ is strongly real in ${\displaystyle G}$.

The extended centralizer of an element ${\displaystyle x}$ of a group ${\displaystyle G}$ is defined as