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 , giving a homomorphism . A basic example would be the cyclic group acting on an -gon by rotation. But here are some more interesting and general group actions for a group :

acts on by left multiplication.

acts on by left multiplication. acts on a normal subgroup of by conjugation.

acts on a normal subgroup of by conjugation. acts on the set of subgroups of of a fixed order, by conjugation.

acts on the set of subgroups of of a fixed order, by conjugation. If , acts on the set of left cosets of by left multiplication.

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

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

We make use of some basic Sylow theory. Let be a simple group of order and let be the number of subgroups of that have order . Sylow’s theorems tell us that is a factor of and is congruent to , so . If , then the group of order is normal, (do you see why?) so .

Let act on the set of subgroups of of order , by conjugation. This gives a homomorphism . Because none of these subgroups can be normal (or again appealing to Sylow theory) it is easy to see that , so . But is a normal subgroup of , so and is injective.

So we can imagine as a subgroup of . In fact, , as the following lemma will tell us:

Lemma: If is a simple group of order larger than , .

Proof: Let be the sign homomorphism, so . Then is a normal subgroup of . But cannot be injective, as , so , in other words .

Returning to our solution, we can now assume that . Counting orders, we see that has index . Let act on the set of left cosets of by left multiplication, giving a homomorphism . Pausing for a moment to verify that , the fact that is simple tells us that is injective, and applying the lemma tells us easily that is an isomorphism.

What is the image of under this isomorphism? Well, let’s think about the action again. We have some cosets . Multiplication on the left by elements of is certainly not guaranteed to fix the first five, but it will always fix the last one. In other words, . So .

Since , it follows that is an isomorphism carrying to , 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.

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