The Unapologetic Mathematician

Mathematics for the interested outsider

Generators and Relations

Now it’s time for the reason why free groups are so amazingly useful. Let X be any set, F(X) be the free group on X, and G be any other group. Now, every function f from X into G extends to a unique homomorphism f: F(X)\rightarrow G. Just write down any word in F(X), send each letter into G like the function tells you, and multiply together the result!

So what does this get us? Well, for one thing every group G is (isomorphic to) a quotient of a free group. If nothing else, consider the free group F(G) on the set of G itself. Then send each element to itself. This extends to a homomorphism f from F(G) to G whose image is clearly all of G. Then the First Isomorphism Theorem tells us that G is isomorphic to F(G)/{\rm Ker}(f). That’s pretty inefficient, but it shows that we can write G like that if we want to. How can we do better?
Read more »

February 24, 2007 Posted by John Armstrong | Algebra, Group theory, Structure of Groups | | 4 Comments

A few more facts about group actions

There’s another thing I should have mentioned before. When a group G acts on a set S, there is a bijection between the orbit of a point x and the set of cosets of G_x in G. In fact, gx=hx if and only if h^{-1}gx=x if and only if h^{-1}g is in G_x if and only if gG_x=hG_x. This is the the bijection we need.

This has a few immediate corollaries. Yesterday, I mentioned the normalizer N_G(K) of a subgroup K. When a subgroup H acts on G by conjugation we call the isotropy group of an element x of G the “centralizer” C_H(x) of x in H. This gives us the following special cases of the above theorem:

  • The number of elements in the conjugacy class of x in G is the number of cosets of C_G(x) in G.
  • The number of subgroups conjugate to K in G is the number of cosets of N_G(K) in G.

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 H is a subgroup of a group G, the number of cosets of H in G is written \left[G:H\right]. Note that this doesn’t have to be a finite number, but when G (and thus H) is finite, it is equal to the number of elements in G divided by the number in H. Also notice that if H is normal, there are \left[G:H\right] elements in G/H.

This is why we could calculate the number of permutations with a given cycle type the way we did: we picked a representative g of the conjugacy class and calculated \left[S_n:C_{S_n}(g)\right].

One last application: We call a group action “free” if every element other than the identity has no fixed points. In this case, G_x is always the trivial group, so the number of points in the orbit of x is \left[G:G_x\right] is the number of elements of G. 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.

February 24, 2007 Posted by John Armstrong | Algebra, Group Actions, Group theory | | 2 Comments