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?
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February 24, 2007 Posted by | Algebra, Group theory, Structure of Groups | 5 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 | Algebra, Group Actions, Group theory | 5 Comments