Now it’s time for the reason why free groups are so amazingly useful. Let be any set, be the free group on , and be any other group. Now, every function from into extends to a unique homomorphism . Just write down any word in , send each letter into like the function tells you, and multiply together the result!
So what does this get us? Well, for one thing every group is (isomorphic to) a quotient of a free group. If nothing else, consider the free group on the set of itself. Then send each element to itself. This extends to a homomorphism from to whose image is clearly all of . Then the First Isomorphism Theorem tells us that is isomorphic to . That’s pretty inefficient, but it shows that we can write like that if we want to. How can we do better?
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.