Groups Lecture 15

in which we meet conjugation in several forms, and learn that lecturers are not infallible.

Today we continued to look at some standard actions. First we saw the left coset action: for a subgroup H\leq G, G acts on the left cosets of H by left multiplication, transitively. Perhaps the most important of our standard actions is the conjugation action: G acts on itself by conjugation, meaning g(x)=gxg^{-1} for g,x\in G. The kernel of this action is the centre of G, all elements which commute with everything. The orbits are the conjugacy classes, and the stabilisers are called centralisers: the stabiliser of an element a\in G is the set of elements which commutes with a. So the intersection of all centralisers gives the centre of G. This conjugation action restricts to normal subgroups of G, because when conjugating an element k\in K by any element g\in G, we land in the normal subgroup K again. We proved also that normal subgroups are exactly those subgroups which are unions of conjugacy classes.

Conjugation doesn’t just work on elements, but also on subgroups. If H\leq G then gHg^{-1} is also a subgroup of G, and so we can show that G acts by conjugation on the set of all subgroups of G. The normal subgroups then are exactly the ones which have singleton ccls. The stabilisers of this action are called normalisers: N_G(H)=\{g\in G\mid gHg^{-1}=H\}. I left it to you to show that the normaliser is in fact the largest subgroup of G in which H is normal.

We started with some applications of actions in general and the Orbit-Stabiliser Theorem in particular. By letting the rotations of a cube act on the set of vertices, we found that the group of rotations has size 24=8\cdot 3: 8 vertices, each with an (easy to find) stabiliser of size 3. We then saw that if we try and do the same thing by looking at the action on the three “axes” or “pairs of opposite faces” that we had last time, finding the stabiliser is much harder to get right.

Understanding today’s lecture

Conjugation is a very important concept. You will find it “hidden” throughout the groups lecture notes so far, and also in other courses. See if you can find out all instances of conjugation in the groups notes!

As an undergrad I used to get confused by all these different names “stabilisers” and “centralisers” and “normalisers”…. It helps if you make yourself a table or something to say “This action has that as stabilisers and that as orbits, this action has these, and they have a special name,” etc.

Main lesson from today: use straight-forward actions to determine the sizes of groups via Orbit-Stabiliser, and then you can use Orbit-Stabiliser to help you find stabilisers of more complicated actions. I think I mentioned this last time already. If we know how many elements we are looking for, it is easier to know whether we’ve got them all or are missing some. In this case I missed some: if you rotate by 180° in an axis through the centre of two opposite edges, that swaps two of the face-axes and fixes one (turning it on its head). So we will have to correct an example from last time as well, when I gave you a stabiliser of that particular action. (Of course I did all this on purpose to make you get the point ;-). )

Preparing for Lecture 16

Next time we will correct the rotations of the cube mistake. So if you haven’t made your own “Taylor cube” yet, now would be a good time, and you could even bring it to the next lecture (it makes it so much clearer!). We will also use the left coset action, so remind yourself of that. You can also read up on Cauchy’s Theorem: there is an element of order p for any prime dividing the order of the group. That will be a longish proof, so reading ahead might help. There are several different versions of proof.

Going a little deeper

Conjugation is a very useful concept. It is used in many areas of mathematics to “make things easier”. For example, at the end of the course we will be talking about Möbius maps, isometries of the Riemann Sphere. For many properties of these maps it is enough to say “it is conjugate to one of these few examples, and we know the properties of those examples very well”. You have the same in Vectors and Matrices, which will become more obvious in Linear Algebra next year: square matrices represent linear maps V\to V with respect to a certain basis, and conjugation of matrices is change of basis. You will see in V+M for 2\times 2 matrices over \mathbb{C}, and in Linear Algebra for general n\times n matrices over \mathbb{C}, that there are certain very nice representatives in each conjugacy class, called the Jordan Normal Form. In two dimensions they come in three types: either diagonal with two different entries on the diagonal, or diagonal with the same entry on the diagonal, or the same entry on the diagonal, a 1 in the right upper corner and a 0 in the left lower corner. We will mention them in this course as well, in the Matrix Groups chapter.

When I was in Part III, I did a lecture course called “Classical Dynamics”, which had some chaos theory and Julia-sets and so on in it; very exciting. We used conjugation over and over again to understand properties of different isometries and so on.

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