in which we explore conjugacy classes in and , and prove that is simple.
Having proved last time that the conjugacy classes in are determined by cycle type, we listed all sizes of the ccls of , and also the sizes of their centralisers, using Orbit-Stabiliser. Knowing the ccls and their sizes, we could have a look at what normal subgroups we could find, because normal subgroups must be unions of ccls. In we found four such, and also determined the corresponding quotients. Here our “groups of order 6” result came in useful. I left you to try the same for .
We then looked at ccls in . It is not quite the same there, as some ccls can split. We saw, using Orbit-Stabiliser for both and , that the ccl of splits in if and only if commutes with no odd permutation in . So we just have two possible situations for any : since , we either have and the same ccls, or we have and the ccl splits. We explored this in and , and in both cases found only one ccl which split. We used the sizes of the ccls also to prove that is simple: the only unions of ccls (including ) which give a size dividing are and all of .
Understanding today’s lecture
Getting your head round the splitting ccls can take a few tries. I remember my lecturer explaining it with “ has index 2 in , and has index 1 or 2 in , so the only possibilities are these.” I didn’t really understand it with the index, it helped me to write down both Orbit-Stabiliser equations explicitly, and then I saw what was going on. So if you don’t understand my explanation, see if you can reformulate it in a way that you like better. Or look in a book, or talk to your colleagues, or your supervisor, or come and ask me :-).
Preparing for Lecture 18
Next time we will finish the promised “groups up to order 8” by looking at the missing order 8. We will see that there are quite a few, which is in fact because 8 is a power of 2. Apart from the obvious products of cyclic groups, we also have a dihedral one and a “new” one, called the Quaternions. We will look at those in a bit more detail. We will then start on matrix groups. We’ve seen a few already throughout, so you could look back to find matrix groups in your lecture notes as a preparation.
Going a little deeper
We saw in the examples of and that only one of the ccls split, that of the largest possible cycle (which is actually in ). You could investigate what happens in general for . Will it always be just the largest possible cycle? Or could there be others as well? I think I started investigating this once, but I can’t quite remember how far I got, so I’d be interested to hear your answers! (Maybe not on the blog, so you don’t spoil it for others who want to try. Or at least with sufficient spoiler alert.)