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 $\mathrm{T}\mathrm{r}(\rho (g))$ 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 $|C_G(x)|$, 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 $|C_G(x)|$ 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

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

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

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,

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

See also

• Brauer–Fowler theorem

References

• Gorenstein, Daniel (2007). Finite Groups. AMS Chelsea Publishing. ISBN 978-0821843420. 
• Isaacs, I. Martin (1994). Character Theory of Finite Groups. Dover Publications. ISBN 978-0486680149. 
• Rose, John S. (2012). A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8.