in which we apply group theory to prove Fermat-Euler, and use Lagrange to help us find subgroups or determine what a small group must look like.
We started by looking for a subset of that behaves well with multiplication. We found that the set of units (elements that have a multiplicative inverse) forms a group under multiplication . This uses the exciting fact that multiplication by gives an injective function from to itself, and being a finite set, it is automatically also surjective, which gives us the existence of inverses.
This let us 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.
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 will lead us, in the next chapter “Quotients”, to define normal subgroups: see more next time.
Understanding today’s lecture
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 (this is relevant on Sheet 3) “the subgroup of rotations” and “any (or a) subgroup of rotations”?
Do you know a different proof for Fermat’s little theorem? How does it compare to this groups proof? Remember that all the work is in showing that is a group. Can you see the connection between “having an inverse under multiplication mod ” and “being coprime to ”? At which point did we make use of this?
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.
Preparing for Lecture 11
Next time we will introduce normal subgroups (we have mentioned them before), and 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 a book about quotient groups (also called factor groups) before next time.
Going a little deeper
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 !