Fun With Group Actions

I had to take a bit of a break to move into my new apartment in the Castro, (which has, incidentally, been a pretty exciting place to live this past week) but now I’m all settled down and ready to tell you about group actions.

The basic idea of a group action is to visualize a group as a set of permutations of some set X, giving a homomorphism \phi: G\to S_{|X|}. A basic example would be the cyclic group Z_n acting on an n-gon by rotation. But here are some more interesting and general group actions for a group G:

  • G acts on G by left multiplication.
  • G acts on a normal subgroup of G by conjugation.
  • G acts on the set of subgroups of G of a fixed order, by conjugation.
  • If H\leq G, G acts on the set of left cosets of H by left multiplication.

All of these can provide useful information through the existence of homomorphisms to some symmetric group S_n. For an example of this technique, let’s consider this classic problem:

 

Problem: Show that any simple group of order 60 is isomorphic to A_5. (recall that a simple group is one with no nontrivial normal subgroups)

We make use of some basic Sylow theory. Let G be a simple group of order 60 and let k be the number of subgroups of G that have order 5. Sylow’s theorems tell us that k is a factor of 12 and is congruent to 1\ (\rm{mod}\ 5), so k\in \{1,6\}. If k=1, then the group of order 5 is normal, (do you see why?) so k=6.

Let G act on the set of subgroups of G of order 5, by conjugation. This gives a homomorphism \phi: G\to S_6. Because none of these subgroups can be normal (or again appealing to Sylow theory) it is easy to see that \phi\neq 1, so \ker{\phi}\neq G. But \ker{\phi} is a normal subgroup of G, so \ker{\phi}=1 and \phi is injective.

So we can imagine G as a subgroup of S_6. In fact, \phi(G)\leq A_6, as the following lemma will tell us:

 

Lemma: If H\leq S_n is a simple group of order larger than 2, H\leq A_n.

Proof: Let \sigma: S_n\to Z_2 be the sign homomorphism, so A_n = \ker{\sigma}. Then \ker{\sigma_H} is a normal subgroup of H. But \sigma_H cannot be injective, as |H|>|Z_2|, so \ker{\sigma_H} = H, in other words H\leq \ker{\sigma}=A_n.

 

Returning to our solution, we can now assume that G\leq A_6. Counting orders, we see that G has index 360/60=6. Let A_6 act on the set of left cosets of G by left multiplication, giving a homomorphism \psi: A_6 \to S_6. Pausing for a moment to verify that \psi\neq 1, the fact that A_6 is simple tells us that \psi is injective, and applying the lemma tells us easily that \psi:A_6\to A_6 is an isomorphism.

What is the image of G under this isomorphism? Well, let’s think about the action again. We have some cosets x_1 G, x_2 G, x_3 G, x_4 G, x_5 G, G. Multiplication on the left by elements of G is certainly not guaranteed to fix the first five, but it will always fix the last one. In other words, \psi(G)\leq S_5. So \psi(G)\leq S_5 \cap A_6=A_5.

Since |G|=|A_5|, it follows that \psi is an isomorphism carrying G to A_5, and we are done.

 

As we can see, group actions provide an excellent if somewhat magical way of reasoning about finite groups that are too big to be easily understood by hand. Make them part of your group theory toolbox.

This entry was posted on Monday, November 10th, 2008 at 3:39 am and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Leave a Reply