Groups Lecture 14

in which we meet several standard actions and prove Cayley’s Theorem.

Having proved the Orbit-Stabiliser Theorem last time, we started today with some examples. The Orbit-Stabiliser Theorem tells us that if an action is transitive, all stabilisers have the same size. But in the example of $\langle (12)\rangle$ acting on $\{1,2,3\}$, we saw that while we always get the same product, the orbits (and stabilisers) need not all have the same size. The Orbit-Stabiliser Theorem is also very useful in helping us find for example stabilisers, because it tells us how big they are to begin with. This helped us prove that the stabiliser of $1$ in $S_4$ really is $S_{\{2,3,4\}}\cong S_3$.

Equipped with Orbit-Stabiliser, we started looking at some standard actions. We investigated the left regular action of $G$ on itself, by left multiplication. This is faithful and transitive. It also implies Cayley’s Theorem: every group is isomorphic to a subgroup of some symmetric group.  We then 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.

Understanding today’s lecture

You will have to get used to the standard actions, they will keep cropping up. In particular, 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!

Conjugation is used in a lot of areas of maths to “look at something in an easier situation”. For example rotations in the complex plane $\mathbb{C}$: if you want to rotate around any point, instead you can first translate that point to $0$, then rotate around $0$ (very easy), and then translate back to the original point. Conjugations to easier cases like this are extremely helpful.

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.

Preparing for Lecture 15

We will see some applications of actions, for example we will work more on the rotations of the cube. 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. 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

We proved today that every group is isomorphic to a subgroup of some symmetric group. In the proof we in fact saw that $G$ is isomorphic to a subgroup of $\mathrm{Sym}G$. Now that is in general  a very very large group. Many groups are subgroups of much smaller symmetry groups. The most obvious one being $S_n$: it is the (whole) group of symmetries on $n$ elements, whereas our proof of Cayley would give it to us as a subgroup of $\mathrm{Sym}S_n=S_{n!}$, which is much bigger. And we’ve seen that the dihedral group $D_{2n}$ can be viewed as a subgroup of $S_n$, rather than as a subgroup of $S_{2n}$ which Cayley would give us. So Cayley doesn’t give us the best way of seeing $G$ as symmetries, but it does show there is always at least one!

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.