Groups Lecture 16

in which we prove Cauchy’s theorem and revisit the symmetric groups.

We started with a correction to an example from Lecture 14.

Today we saw how the left coset action can help us find out things about possible subgroups of a group G. If H\leq G of index n, through the left coset action we get a group homomorphism \varphi\colon {G\to S_n}. The kernel of this group hom is a normal subgroup of G. This can now be used in different ways for different situations. If H\neq G, then we know the kernel is not all of G, so either we find a normal subgroup (which could be \{e\}), or if we know that G is simple, then the kernel must be \{e\}. Then we have an injection G\to S_n, which gives us a restriction on n through the size of G: |G|\leq n!. Using the signature homomorphism as well lets us refine this to |G|\leq \frac{n!}{2} for simple groups.

Our main theorem today was Cauchy’s Theorem: every finite group G with prime p dividing |G| contains an element of order p. We saw earlier in the course that this is not the case for non-primes! There are several different proofs of this theorem, but we saw this one: we acted with a cyclic group of order p on the set of p-tuples of G which multiply to the identity. By looking at the size of orbits, which partition the set we are acting on, we found that we must have some orbits of size 1 (in addition to the one containing (e,e,\ldots,e)). And such a singleton orbit gives us an element of order p.

We then revisited symmetric groups and looked at their conjugacy classes. In S_n, conjugacy classes are determined by cycle type.

Understanding today’s lecture

The argument using left coset action is a very useful one! So when you get to do example sheet questions, remember it :-). The proof of Cauchy’s Theorem can look a bit daunting at first. But perhaps my summary above will help to make it a little clearer. You can also write your own summary of it. There are different proofs, so if you really don’t like this one, see if you can find another one in a book. I chose this one because it just uses one action rather than lots of “magic” combinations of different actions pulled from thin air.

The ccls in S_n are something people usually get used to very quickly. Just be careful you don’t get too used to it, it doesn’t quite work like that in A_n! So never switch off your brain.

Preparing for Lecture 17

In the next lecture we will see what the ccls sizes of S_4 tell us about its normal subgroups and so its quotients. We will investigate how ccls work in A_n. They are not quite the same there: some ccls from S_n can split when we go to A_n. We will look at ccls of A_4 and A_5 and use them to prove that A_5 is simple.

Going a little deeper

The left coset action argument is a nice example of how to use theory rather than do everything by hand from first principles. As a category theorist, I much prefer using general results and structure to messing around with details :-). Though both methods have their place. If you want to get a deeper understanding, you should train yourself to look for connections between different results and situations though. In this argument we used the Alternative Action Definition, we used that every kernel is normal, we later used the signature homomorphism as well. Putting together these results from different chapters/subareas gives us more insight into the situation we are faced with.

If you look back through your lecture notes, you might (or should) find one or two places where I would have loved to use Cauchy’s Theorem but couldn’t because we hadn’t proved it yet. So we had to find another explanation. Can you find at least one such a place? Also, if you look back at “exponent divides group order”, we can now see that in fact the exponent has every prime as a factor that is a prime factor of |G|. Can you prove it?

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