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 $G$ {{nowrap begin}}with $x^g = x^{-1}$ ,{{nowrap end}} where $x^g$ is defined as $g^{-1} \cdot x \cdot g$ .{{sfnp|Rose|2012|p=111}} An element $x$ of a group $G$ is called strongly real if there is an involution $t$ with {{nowrap begin}} $x^t = x^{-1}$ .{{nowrap end}}{{sfnp|Rose|2012|p=112}}

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 $G$ .{{sfnp|Isaacs|1994|p=31}}

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.{{sfnp|Isaacs|1994|p=31}}

For a real element $x$ of a group $G$ , the number of group elements $g$ {{nowrap begin}}with $x^g = x^{-1}$ {{nowrap end}} is equal to $\left|C_G(x)\right|$ ,{{sfnp|Rose|2012|p=111}} 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 {{nowrap| $x \ne e$ }} 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 {{nowrap| $\left\{x, x^{-1}\right\}$ .}}{{sfnp|Rose|2012|p=86}}

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.{{sfnp|Rose|2012|p=111}} 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

References

{{cite book |last1=Gorenstein |first1=Daniel |author1-link=Daniel Gorenstein |title=Finite Groups |publisher=AMS Chelsea Publishing |isbn=978-0821843420 |year=2007 |orig-year=reprint of a work originally published in 1980}}
{{cite book |last=Isaacs |first=I. Martin |author-link=Martin Isaacs |title=Character Theory of Finite Groups |publisher=Dover Publications |isbn=978-0486680149 |year=1994 |orig-year=unabridged, corrected republication of the work first published by Academic Press, New York in 1976 }}
{{cite book |last=Rose |first=John S. |date=2012 |title=A Course on Group Theory |publisher=Dover Publications |isbn=978-0-486-68194-8 |orig-year=unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978 }}

Category:Group theory

Real element

Properties

Extended centralizer

See also

Notes

References