Real element

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 ${\displaystyle g^{-1}\cdot x\cdot g}$.^[1] An element ${\displaystyle x}$ of a group ${\displaystyle G}$ is called strongly real if there is an involution ${\displaystyle t}$ with ${\displaystyle x^{t}=x^{-1}}$.^[2]

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 ${\displaystyle G}$.^[3]

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.

Properties

A group with real elements other than the identity element necessarily is of even order.^[3]

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 ${\displaystyle \left|C_{G}(x)\right|}$,^[1] where ${\displaystyle C_{G}(x)}$ is the centralizer of ${\displaystyle x}$,

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

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

Extended centralizer

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

The extended centralizer of an element of a group ${\displaystyle G}$ is always a subgroup of ${\displaystyle G}$. For involutions or non-real elements, centralizer and extended centralizer are equal.^[1] For a real element ${\displaystyle x}$ of a group ${\displaystyle G}$ that is not an involution,

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

See also

• Brauer–Fowler theorem

Notes

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 978-0-486-68194-8.