In group theory, a discipline within modern algebra, an element <math>x</math> of a group <math>G</math> is called a '''real element''' of <math>G</math> if it belongs to the same conjugacy class as its inverse <math>x^{-1}</math>, that is, if there is a <math>g</math> in <math>G</math> {{nowrap begin}}with <math>x^g = x^{-1}</math>,{{nowrap end}} where <math>x^g</math> is defined as <math>g^{-1} \cdot x \cdot g</math>.{{sfnp|Rose|2012|p=111}} An element <math>x</math> of a group <math>G</math> is called '''strongly real''' if there is an involution <math>t</math> with {{nowrap begin}}<math>x^t = x^{-1}</math>.{{nowrap end}}{{sfnp|Rose|2012|p=112}}
An element <math>x</math> of a group <math>G</math> is real if and only if for all representations <math>\rho</math> of <math>G</math>, the trace <math>\mathrm{Tr}(\rho(g))</math> of the corresponding matrix is a real number. In other words, an element <math>x</math> of a group <math>G</math> is real if and only if <math>\chi(x)</math> is a real number for all characters <math>\chi</math> of <math>G</math>.{{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 <math>S_n</math> of any degree <math>n</math> 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 <math>x</math> of a group <math>G</math>, the number of group elements <math>g</math> {{nowrap begin}}with <math>x^g = x^{-1}</math>{{nowrap end}} is equal to <math>\left|C_G(x)\right|</math>,{{sfnp|Rose|2012|p=111}} where <math>C_G(x)</math> is the centralizer of <math>x</math>,
:<math>\mathrm{C}_G(x) = \{ g \in G\mid x^g = x \}</math>.
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|<math> x \ne e</math>}} and <math>x</math> is real in <math>G</math> and <math>\left|C_G(x)\right|</math> is odd, then <math>x</math> is strongly real in <math>G</math>.
== Extended centralizer == The '''extended centralizer''' of an element <math>x</math> of a group <math>G</math> is defined as
:<math>\mathrm{C}^*_G(x) = \{ g \in G\mid x^g = x \lor x^g = x^{-1} \},</math>
making the extended centralizer of an element <math>x</math> equal to the normalizer of the set {{nowrap|<math>\left\{x, x^{-1}\right\}</math>.}}{{sfnp|Rose|2012|p=86}}
The extended centralizer of an element of a group <math>G</math> is always a subgroup of <math>G</math>. For involutions or non-real elements, centralizer and extended centralizer are equal.{{sfnp|Rose|2012|p=111}} For a real element <math>x</math> of a group <math>G</math> that is not an involution,
:<math>\left|\mathrm{C}^*_G(x):\mathrm{C}_G(x)\right| = 2.</math>
==See also== * Brauer–Fowler theorem
==Notes== {{reflist}}
==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