A few more facts about group actions
There’s another thing I should have mentioned before. When a group acts on a set , there is a bijection between the orbit of a point and the set of cosets of in . In fact, if and only if if and only if is in if and only if . This is the the bijection we need.
This has a few immediate corollaries. Yesterday, I mentioned the normalizer of a subgroup . When a subgroup acts on by conjugation we call the isotropy group of an element of the “centralizer” of in . This gives us the following special cases of the above theorem:
- The number of elements in the conjugacy class of in is the number of cosets of in .
- The number of subgroups conjugate to in is the number of cosets of in .
In fact, since we’re starting to use this “the number of cosets” phrase a lot it’s time to introduce a bit more notation. When is a subgroup of a group , the number of cosets of in is written . Note that this doesn’t have to be a finite number, but when (and thus ) is finite, it is equal to the number of elements in divided by the number in . Also notice that if is normal, there are elements in .
This is why we could calculate the number of permutations with a given cycle type the way we did: we picked a representative of the conjugacy class and calculated .
One last application: We call a group action “free” if every element other than the identity has no fixed points. In this case, is always the trivial group, so the number of points in the orbit of is is the number of elements of . We saw such a free action of Rubik’s Group, which is why every orbit of the group in the set of states of the cube has the same size.