View source for Conjugate (group theory)

Let <math>G</math> be a [[group]] operating on a [[set]] <math>S</math>.  An element <math>y\in S</math> ''conjugate'' to an element <math>x\in S</math> if there exists an element <math>\alpha \in G</math> such that <math>y = \alpha x</math>.  The relation of conjugacy is an [[equivalence relation]].  The set of conjugates of an element <math>x</math> of <math>S</math> is called the [[orbit]] of <math>x</math>.

Note that this definition conforms to the notion of [[complex conjugate]].  Indeed, under the group of [[field]] [[automorphism]]s on the complexe numbers that do not change the reals, the orbit of a complex number <math>z</math> is the set <math>\{z, \bar{z}\}</math>.

If <math>H</math> is a [[subset]] of a group <math>G</math>, the '''conjugate''' of <math>H</math> usually means the conjugate of <math>H</math> under the group of [[inner automorphism]]s acting on the subsets of <math>G</math>.  If <math>H</math> is a [[subgroup]] of <math>G</math>, any conjugate of <math>H</math> is also a subgroup, as for any <math>a \in G</math>,
<cmath> (aHa^{-1})(aHa^{-1}) = aHHa^{-1} = aHa^{-1}. </cmath>

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== See also ==

* [[Orbit]]
* [[Stabilizer]]

[[Category:Group theory]]