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
Famous quotes containing the words group and/or action:
“Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
—Mary Barnett Gilson (1877–?)
“Therefore all just persons are satisfied with their own praise. They refuse to explain themselves, and are content that new actions should do them that office. They believe that we communicate without speech, and above speech, and that no right action of ours is quite unaffecting to our friends, at whatever distance; for the influence of action is not to be measured by miles.”
—Ralph Waldo Emerson (1803–1882)