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

An element $x$ of a group $G$ is real if and only if for all representations $\rho$ of $G$ , the trace $\mathrm {Tr} (\rho (g))$ 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 $G$ .^[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.^[3]

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

\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 $\left\{x,x^{-1}\right\}$ .^[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,

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

Notes

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

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Real element

Properties

Extended centralizer

See also

Notes

References