In which we explore the Orbit-Stabiliser Theorem.
- Proposition 57: Let be a group acting on a set . Take . Then is a subgroup of . This was a quick check using the definition of the stabiliser and of the group action.
- Proposition 58: Let be a group acting on a set . Take , with and lying in the same orbit. Then and are conjugate: there is with . We noted that since and are in the same orbit, there is with . And then we showed that if and only if .
- Theorem 59 (Orbit-Stabiliser Theorem): Let be a finite group acting on a set . Take . Then . We defined a map from to and showed that it’s a bijection, then used Lagrange’s theorem.
- Corollary 60: Let be a finite group, take . Then , where is the centraliser of in and is the conjugacy class of in . We have already seen that acts on itself by conjugation, and that for we have and , so the result follows immediately from Orbit-Stabiliser.
- Proposition 61: Let be prime. Let be a group with order for some . Then the centre of is non-trivial (contains an element other than ). We used the action of on itself by conjugation. The elements of the centre are precisely the elements whose orbits have size 1. The size of each orbit divides and so is a multiple of , and since the orbits partition we see that the sum of their sizes is a multiple of . So The number of orbits of size 1 is a multiple of , and since it’s at least 1 it must be at least .
Understanding today’s lecture
By thinking of the group of rotational symmetries of the cube as acting on the edges of the cube, can you show that the group has size 24? What is the size of the group of rotational symmetries of the tetrahedron? Of the dodecahedron?
And here’s a dodecahedron I made earlier (using instructions from NRICH), in case you’re having difficulties visualising one.
Here’s Tim Gowers on the orbit-stabiliser theorem. And there are some useful summary notes here from Thomas Beatty. These lecture notes by Keith Carne show how one can start with orbit-stabiliser and get to all sorts of exciting mathematics.
Preparation for Lecture 14
What can you say about the centre of a group of order (where is prime)?
What can you say about the structure of a group of order ?