Difference between revisions of "Conjugacy class"
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| − | A '''conjugacy class''' is | + | A '''conjugacy class''' is a particular subset of a [[group]]. |
| − | Let <math>G</math> be a group. Consider the action of <math>G</math> on itself by [[inner automorphism]]s. The [[orbit]]s of <math>G</math> are then called '''conjugacy classes'''. | + | Let <math>G</math> be a group. Consider the action of <math>G</math> on itself by [[inner automorphism]]s. The [[orbit]]s of <math>G</math> are then called '''conjugacy classes'''. By expanding the definition, it is easy to show that two elements <math>g</math> and <math>g'</math> are in the same conjugacy class iff there is an element <math>x</math> such that <math>g' = x^{-1}gx</math>. |
Two [[subset]]s <math>H</math> and <math>H'</math> of <math>G</math> are called ''conjugate'' if there exists <math>\alpha \in G</math> for which <math>H</math> is the image of <math>H'</math> under <math>\text{Int}(\alpha)</math>. | Two [[subset]]s <math>H</math> and <math>H'</math> of <math>G</math> are called ''conjugate'' if there exists <math>\alpha \in G</math> for which <math>H</math> is the image of <math>H'</math> under <math>\text{Int}(\alpha)</math>. | ||
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| + | The [[characters|character]] of any group <math>G</math> are constant on conjugacy classes. | ||
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Latest revision as of 11:00, 16 July 2020
A conjugacy class is a particular subset of a group.
Let 
be a group. Consider the action of on itself by inner automorphisms. The orbits of are then called conjugacy classes. By expanding the definition, it is easy to show that two elements 
and 
are in the same conjugacy class iff there is an element 
such that 
.
Two subsets 
and 
of are called conjugate if there exists 
for which is the image of under 
.
The character of any group are constant on conjugacy classes.
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