in which we apply group theory to prove Fermat-Euler, use Lagrange to help us find subgroups or determine what a small group must look like, and meet normal subgroups.
Remember that last time we found that the set of units (elements that have a multiplicative inverse) forms a group under multiplication .
This allowed us to prove Fermat-Euler: for a natural number and an integer coprime to , we have . The is Euler’s totient function “the number of elements in which are coprime to “. It doesn’t look like it has anything to do with groups at first, but we proved it very easily using the fact that is a group under multiplication of size and that the exponent of a group divides the order of the group. A special case of this is Fermat’s Little Theorem, which is the case when is a prime, and so .
As further applications of Lagrange, we found all the subgroups of and , Lagrange helping significantly in our search by telling us what sizes to look for. We also proved that any group of order is (isomorphic to) either or . That means (up to isomorphism) there are only two groups of order ! Quite a useful fact. See “going deeper” to see what it has to do with turning over mattresses.
We then looked at left and right cosets again and saw that sometimes they are the same and sometimes they are not. This lead us to define normal subgroups, which brings us to the start of the fourth chapter, “Quotient groups”. Normal subgroups are those for which each left coset is the same as the right coset . We can also say: for any and , we have . We will see later that this could be called “stable under conjugation” or something like that. We showed that subgroups of index and subgroups in abelian groups are always normal. Next time we will see that kernels are also always normal.
Understanding today’s lecture
Look through how we found the subgroups of and . Where were we completely guided and had no options? Where did we actually have to look round to see what we could find? Use Lagrange to help you find subgroups of a few other groups. I suggested in lectures, which is really helpful, you get to know a lot about cyclic groups while doing it.
Normal subgroups are very important. Make sure you are happy with the equivalence of the three different ways of “defining” or checking whether something is a normal subgroup. You will probably find different ones useful in different contexts. Go through all the subgroups you’ve met so far (for example in the earlier lectures, or on the practice sheets, and on the example sheets) and find out which ones are normal.
Remember that being able to distinguish your capital Ks from your small case ks is crucial to the understanding of normal subgroups!
Preparing for Lecture 11
We will look at kernels next time, so look up the definition of those if you don’t remember. And as promised, we will find out what groups of size there are. This uses Lagrange and normal subgroups: perhaps you can have a go already yourself.
We will also start working on quotients, which from experience a lot of students don’t find so easy. My advice is to look at it as often as you can, as it will get easier with each time you see it. So perhaps read ahead in the lecture notes about quotient groups (or in a book) before next time.
Going a little deeper
Here are some language subtlety that we expect you to just pick up (probably harder for people who’s first language is not English): what do you think is the difference between saying “these are units” and “these are the units”? Or “the set of units” and “a set of units”? Or (this is relevant on Sheet 3) “the subgroup of rotations” and “any (or a) subgroup of rotations”? These are actually important mathematical questions which you will meet (in some kind or other) often. You might notice that in lectures I sometimes leave out “fill-words” like “the”, but in such a case one really cannot leave it out.
What do groups of order have to do with turning over mattresses? Well, you may not know this yet (I only learnt it when I was 30 or so), but it is recommended that you turn over your mattresses both head to feet and upside to downside, so that it gets equal wear all over. Now it is quite hard to remember what you did last (as you don’t tend to do it all that frequently), so wouldn’t it be nice if we could have just one way of turning it which automatically cycles through all the four different options? What does this have to do with groups? Well, first think about the fact that turning over a mattress, with “composition”, forms a group. Then we see it has four elements. But if there were one way of doing it which reaches everything, that would mean it is a cyclic group (of order ). But in fact, as you can easily see, all the turns have order , so in fact we get , which is not cyclic. And our theorem tells us that those two are the only options for groups of order !