Difference between revisions of "Conjugate (group theory)"
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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>. | 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, \ | + | 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>. |
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| + | 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> | ||
{{stub}} | {{stub}} | ||
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== See also == | == See also == | ||
| + | * [[Conjugacy class]] | ||
* [[Orbit]] | * [[Orbit]] | ||
* [[Stabilizer]] | * [[Stabilizer]] | ||
[[Category:Group theory]] | [[Category:Group theory]] | ||
Latest revision as of 21:20, 21 May 2008
Let 
be a group operating on a set 
. An element 
conjugate to an element 
if there exists an element 
such that 
. The relation of conjugacy is an equivalence relation. The set of conjugates of an element 
of is called the orbit of .
Note that this definition conforms to the notion of complex conjugate. Indeed, under the group of field automorphisms on the complexe numbers that do not change the reals, the orbit of a complex number 
is the set 
.
If 
is a subset of a group , the conjugate of usually means the conjugate of under the group of inner automorphisms acting on the subsets of . If is a subgroup of , any conjugate of is also a subgroup, as for any 
,
![\[(aHa^{-1})(aHa^{-1}) = aHHa^{-1} = aHa^{-1}.\]](http://latex.artofproblemsolving.com/d/2/5/d25afafbd08a2f5b9f9d612954266d58a9b70f93.png)
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