Conjugacy As Group Action
If we define
- g . x = gxg−1
for any two elements g and x in G, then we have a group action of G on G. The orbits of this action are the conjugacy classes, and the stabilizer of a given element is the element's centralizer.
Similarly, we can define a group action of G on the set of all subsets of G, by writing
- g . S = gSg−1,
or on the set of the subgroups of G.
Read more about this topic: Conjugacy Class
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