Groups Lecture 13

in which we start learning about group actions.

To finish off the Chapter on Quotients, we used the Isomorphism Theorem to show essential uniqueness of cyclic groups: any cyclic group is isomorphic to either (\mathbb{Z},+) or (\mathbb{Z}/n\mathbb{Z},+_n), for some n\in \mathbb{N}.  We also defined a simple group as one which only has \{e\} and itself as normal subgroups. Examples are C_p for p prime (by Lagrange; in fact they have no other subgroups at all, let alone normal ones), and A_5, the alternating group on 5 elements, which we will investigate more in Chapter 6.

Then we started on Group Actions (Chapter 5). An action of a group G on a set X is a function \theta\colon {G\times X\to X} which satisfies:

0. \theta(g,x)\in X for all g\in G, x\in X;
1. \theta(e,x)=x for all x\in X;
2. \theta(g,\theta(h,x))=\theta(gh,x) for all g,h\in G, x\in X.

We saw several other alternative notations to write this \theta, for example \theta(g,x)=g\cdot x=g(x), or we can view \theta(g, - )\colon {X\to X}, for a fixed g\in G, as a map \theta_g\colon {X \to X}. We proved that these \theta_g are all bijections. As examples for actions we saw the trivial action, which does nothing, S_n and D_{2n} both acting on \{1,2,\ldots, n\}, and rotations of a cube acting on several different sets, such as the faces of the cube, the diagonals, the “pairs of opposite faces”, etc.

Using the fact that each \theta_g is a bijection on X, 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.

Understanding today’s lecture

Actions are what make groups tangible or concrete. Through actions, the groups are really “doing” something. We have had several different notations: see that you can translate between them. And maybe you have a favourite one? In lectures we will use different ones in different contexts, as they all have their advantages and disadvantages, and it depends what you want to do as to which is most useful.

We will talk more about rotations of a cube as well in future lectures. I suggest you make yourself your own “Taylor-cube” to play around with, you can find the net to print out here. You may very well want to check out Gareth Taylor’s website which has many more useful things on it.

Preparing for Lecture 14

Next time we will keep exploring actions, by looking at orbits and stabilisers. Orbits are all elements of the set that can be reached from a given element, and stabilisers are all group elements which fix a particular element. You can read up on these in any group theory book before next time. We will prove that orbits partition the set X, and the very important Orbit-Stabiliser Theorem, which is extremely useful for finding sizes of all sorts of things. The proof is a sort of “geometric version of Lagrange”, so go and freshen up on the proof of Lagrange.

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…).

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