Groups Lecture 17

in which we explore conjugacy classes in S_n and A_n, and prove that A_5 is simple.

Having proved last time that the conjugacy classes in S_n are determined by cycle type, we listed all sizes of the ccls of S_4, 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 S_4 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 S_5.

We then looked at ccls in A_n. It is not quite the same there, as some ccls can split. We saw, using Orbit-Stabiliser for both S_n and A_n, that the ccl of \sigma\in A_n splits in A_n if and only if \sigma commutes with no odd permutation in S_n. So we just have two possible situations for any \sigma\in A_n: since C_{A_n}(\sigma)=C_{S_n}(\sigma)\cap A_n, we either have |C_{S_n}(\sigma)|=2|C_{A_n}(\sigma)| and the same ccls, or we have C_{S_n}(\sigma)=C_{A_n}(\sigma) and the ccl splits. We explored this in A_4 and A_5, and in both cases found only one ccl which split. We used the sizes of the ccls also to prove that A_5 is simple: the only unions of ccls (including e) which give a size dividing |A_5|=60 are \{e\} and all of A_5.

We started on the proof of what the possible groups of order 8 are, and we will finish it next time.

Understanding today’s lecture

Getting your head round the splitting ccls can take a few tries. I remember my lecturer explaining it with “A_n has index 2 in S_n, and C_{A_n}(\sigma) has index 1 or 2 in C_{S_n}(\sigma), 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 ask on the forum on the moodle page :-).

Preparing for Lecture 18

Next time we will finish the proof about groups of 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 A_4 and A_5 that only one of the ccls split, that of the largest possible cycle (of those who are actually in A_n). You could investigate what happens in general for A_n. 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.)

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