Editing Conjugacy class (section)

==Conjugacy as group action==<!-- This section is linked from [[Quaternions and spatial rotation]] -->

For any two elements <math>g, x \in G,</math> let
<math display="block">g \cdot x := gxg^{-1}.</math>
This defines a [[Group action (mathematics)|group action]] of <math>G</math> on <math>G.</math> The [[Group action (mathematics)#Orbits and stabilizers|orbits]] of this action are the conjugacy classes, and the [[Group action (mathematics)#Orbits and stabilizers|stabilizer]] of a given element is the element's [[centralizer]].<ref name="Grillet-2007-p56">Grillet (2007), [{{Google books|plainurl=y|id=LJtyhu8-xYwC|page=56|text=the orbits are the conjugacy classes}} p. 56]</ref>

Similarly, we can define a group action of <math>G</math> on the set of all [[subset]]s of <math>G,</math> by writing
<math display="block">g \cdot S := gSg^{-1},</math>
or on the set of the subgroups of <math>G.</math>