Difference between revisions of "Conjugacy class"

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A '''conjugacy class''' is part of a [[group]].
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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'''.
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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 12:00, 16 July 2020

A conjugacy class is a particular subset of a group.

Let $G$ be a group. Consider the action of $G$ on itself by inner automorphisms. The orbits of $G$ are then called conjugacy classes. By expanding the definition, it is easy to show that two elements $g$ and $g'$ are in the same conjugacy class iff there is an element $x$ such that $g' = x^{-1}gx$.

Two subsets $H$ and $H'$ of $G$ are called conjugate if there exists $\alpha \in G$ for which $H$ is the image of $H'$ under $\text{Int}(\alpha)$.

The character of any group $G$ are constant on conjugacy classes.

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See also

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