Group actions

Okay, now we’ve got all the setup for one big use of group theory.

Most mathematical structures come with some notion of symmetries. We can rotate a regular -sided polygon of a turn, or we can flip it over. We can rearrange the elements of a set. We can apply an automorphism of a group. The common thread in all these is a collection of reversible transformations. Performing one transformation and then the other is certainly also a transformation, so we can use this as a composition. The symmetries of a given structure form a group!

What if we paint one side of the above -gon black and the other white, so flipping it over definitely changes it. Then we can only rotate. The rotations are the collection of symmetries that preserve the extra structure of which side is which, and they form a subgroup of the group of all symmetries of the -gon. The symmetries of a structure preserving some extra structure form a subgroup!

As far as we’re concerned right now, mathematical structures are all built on sets. So the fundamental notion of symmetry is rearranging the elements of a set. Given a set , the set of all bijections from to itself is a group. We’ve actually seen a lot of these before. If is a finite set with elements, is the symmetric group on letters.

Now, a group action on a set is simply this: a homomorphism from to . That is, for each element in there is a bijection of and .

It’s important to note what seems like a switch here. It’s really due to the notation, but can easily seem confusing. What’s written on the right is “do the permutation corresponding to , then the one corresponding to “. So we have to think of the multiplication in as “first do , then do “.

In what follows I’ll often write as . The homomorphism property then reads

I’ll throw out a few definitions now, and follow up with examples in later posts.

We can slice up into subsets so that if and are in the same subset, for some , and not if they’re not in the same subset. In fact, this is rather like how we sliced up itself into cosets of . We call these slices the “orbits” of the action.

As an important special case of the principle that fixing additional structure induces subgroups, consider the “extra structure” of one identified point. We’re given an element of , and want to consider those transformations in which send to itself. Verify that this forms a subgroup of . We call it the “isotropy group” or the “stabilizer” of , and write it .

I’ll leave you to ponder this: if and are in the same -orbit, show that and are isomorphic.