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>. |
+ | |||
+ | 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 22:20, 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
,
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