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 $G$ Template:Nowrap beginwith $x^g=x^{-1}$,Template:Nowrap end where $x^g$ is defined as $g^{-1}\cdot x\cdot g$.Template:Sfnp An element $x$ of a group $G$ is called strongly real if there is an involution $t$ with Template:Nowrap begin$x^t=x^{-1}$.Template:Nowrap endTemplate:Sfnp

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 $G$.Template:Sfnp

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.Template:Sfnp

For a real element $x$ of a group $G$, the number of group elements $g$ Template:Nowrap beginwith $x^g=x^{-1}$Template:Nowrap end is equal to $|C_G(x)|$,Template:Sfnp 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 Template:Nowrap 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 Template:NowrapTemplate:Sfnp

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.Template:Sfnp 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

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