Groups Lecture 14

in which we meet orbits and stabilisers and prove the Orbit-Stabiliser Theorem.

Given an action of G on X, we can look at which elements of X can be “reached” from a particular x\in X. The orbit of x is \mathrm{orb}(x)=G(x)=\{y\in X\mid \exists g\in G \text{ s.t. } g(x)=y\}\subseteq X. The stabiliser of x is \mathrm{stab}(x)=G_x=\{g\in G\mid g(x)=x\}\subseteq G, all the elements of G which keep x fixed. Notice that I tend to write these orb and stab sometimes with lower-case letters and sometimes with capital letters; that has no significance. We proved that the stabilisers are subgroups of G (rather than just subsets), and defined an action as transitive if the orbit of any element is the whole set X. We showed that the orbits of an action partition the set X, and this helped us to prove the Orbit-Stabiliser Theorem: If a finite group G acts on a set X, then for any x\in X we have |G|=|\mathrm{orb}(x)||\mathrm{stab}(x)|. Beardon calls this a “geometric version of Lagrange”, and we proved it by “turning it into Lagrange”: The left cosets g\mathrm{Stab}(x) are the sets which contain all h which take x to g(x)= y\in \mathrm{orb}(x), and there is exactly one for each element y in the orbit of x. 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. Today 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 will have more standard actions next time.

Understanding today’s lecture

Orbits and stabilisers are really fun once you get your teeth into them. And the Orbit-Stabiliser Theorem is so immensely useful! Remember to try and use it whenever you want to find a stabiliser. It is also useful in other situations of course. The standard actions you will have to get used to, they will keep cropping up. We’ve just had one so far, so you have time to play with that one before we get the others.

Preparing for Lecture 15

We will get to know more standard actions next time. The left coset action of a group G on left cosets of a subgroup H\leq G, and the very important conjugation action. Conjugating an element a\in G by an element g\in G means taking the element gag^{-1}\in G. You could think already why this will give an action. We will prove that G acts on itself by conjugation, but there are also other sets that G can act on by conjugation: on any normal subgroup K\trianglelefteq G, or on the set of subgroups of G. 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. Though in fact for this next bit, any cube will do :-). 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. I’m not sure if we will get there next time, but certainly the one after that.

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!

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