Groups Lecture 13

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

Last time we defined group actions, and had lots of different notations, including for each g\in G a map \theta_g=\theta(g,-)\colon X\to X. We started today’s lecture by showing that each \theta_g is a bijection on X. Using this fact, and that 2. says \theta_g\circ \theta_h=\theta_{gh} in this notation, we proved an alternative action definition: \theta\colon {G\times X\to X} is an action if and only if \varphi\colon {G\to \mathrm{Sym}X} defined by \varphi(g)=\theta_g is a group homomorphism. This proof is really just translating the conditions of an action into the different notations. We defined the kernel of an action to be the kernel of this group hom \varphi\colon {G\to \mathrm{Sym}X}. These are all the elements of G which “act as the identity”, that means, they don’t do anything to X. An action with trivial kernel is called faithful. We saw that D_{2n} acts faithfully on \{1,2,\ldots,n\}, but that for example the rotations of the cube acting on the pairs of opposite faces (or the three axes through faces) does have a non-trivial kernel.

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.

Understanding today’s lecture

The “Alternative Action Definition” is a very useful way of looking at actions. In some sense, it tells us that when a group acts on a set, then the group elements really “are” bijections from the set to itself, which also interact in a nice way using the group operation. We will use this alternative definition quite often, so make sure you are happy with it.

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.

Preparing for Lecture 14

Next time we will get to know a handful of standard actions. Every group acts on itself by left multiplication, which gives the “left regular action”. Every group also acts on the left coset of a subgroup, and it also acts on itself by conjugation, which is very important. 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.

Going a little deeper

If you want to play around more with actions of the rotations of a cube on different things, try diagonals (the “Taylor cube” has those on). Or if you want a real challenge, look at rotations of a dodecahedron. You could make Gareth’s “dodecahdron showing five cubes inside it”, and see how the rotations of the dodecahedron act on those five cubes. That is a notorious one that comes up in Groups Rings and Modules often (depending on who lectures it…).

2 thoughts on “Groups Lecture 13”

  1. Thanks for posting this lecture! I have a question regarding the action of D_{2n}. When we say the action of rotations on the corners of a cube, do we need to specify the centre of rotation? Or we just assume the centre of rotation is the geometric centre of the cube?

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    • Well, you seem to be mixing up two things: The group D_{2n} is the dihedral group, the symmetries of an n-gon in the plane, so nothing to do with a cube. Symmetries of an n-gon means that the n-gon has to look the same again afterwards (except that the vertices might be numbered differently). So any rotation in D_{2n} must have centre the centre of the n-gon, otherwise it is not a symmetry of the n-gon. This is why we don’t have to say it all the time: it is the only thing possible.
      If you want to talk about rotations of a cube, then you would say “the rotations of a cube act on the vertices of the cube”, which implies that the rotations are such that the cube looks like a cube again afterwards. So rotational axes have to go through the centre of the cube.
      I hope that helps.

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