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 ${\displaystyle x^{g}=x^{-1}}$, where ${\displaystyle x^{g}}$ is defined as ${\displaystyle g^{-1}\cdot x\cdot g}$.^[1] An element $x$ of a group $G$ is real if and only if $\chi (x)$ is a real number for all characters $\chi$ of $G$.^[2]

An element $x$ of a group $G$ is called strongly real if there is an involution $t$ with ${\displaystyle x^{t}=x^{-1}}$.^[3]

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.^[2]

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

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

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 ${\displaystyle x\neq e}$ and $x$ is real in $G$ and ${\displaystyle \left|C_{G}(x)\right|}$ 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

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

making the extended centralizer of an element $x$ equal to the normalizer of the set ${\displaystyle \left\{x,x^{-1}\right\}}$.^[4]

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.^[1] For a real element $x$ of a group $G$ that is not an involution,

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

See also

• Brauer–Fowler theorem

Notes

1. Rose (2012), p. 111.
2. Isaacs (1994), p. 31.
3. Rose (2012), p. 112.
4. Rose (2012), p. 86.

References

• Gorenstein, Daniel (2007) [reprint of a work originally published in 1980]. Finite Groups. AMS Chelsea Publishing. ISBN 978-0821843420.
• Isaacs, I. Martin (1994) [unabridged, corrected republication of the work first published by Academic Press, New York in 1976]. Character Theory of Finite Groups. Dover Publications. ISBN 978-0486680149.
• Rose, John S. (2012) [unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978]. A Course on Group Theory. Dover Publications. ISBN 0-486-68194-7.