Available online
Description
This programme establishes and develops the concept of group action.
This programme establishes and develops the concept of group action.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 24 |
| First transmission date: | 1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:16 |
| + Show more... | |
| Producer: | Martin Wright |
| Contributors: | David Asche; Ian Stewart |
| Publisher: | BBC Open University |
| Keyword(s): | Axioms; Computer animation; Conjugation; Euclid; Non-euclidean geometry; Plane geometry |
| Footage description: | The concept of a group action is established in terms of an abstract group acting on an abstract set. The axioms for a group action are then presented to the viewer, being loosely justified along the lines that a group action should in some way mirror the group. A specific example of a group is considered - the real numbers under addition acting on points in the plane by rotation. This example is checked against the axioms and then the idea of an orbit and stabilizer of a point in the set is introduced. A finite group is now considered - the dihedral group of order eight - the symmetry group of the square. The orbits and stabilizers of various points in the square are considered, and the observation is made that for this finite group it would seem as though the product of the order of the stabilizer and the size of the orbit is the order of the group. But why? By considering one point in the square it is argued that cosets of the stabilizer of that point have a 1-1 correspondence to points in its orbit - which is the orbit-stabilizer theorem the programme seeks to establish. The viewer is referred to the text for a general proof. Having established the orbit stabilizer theorem, the product relationship between the size of orbit and order of stabilizer is quickly justified. A computer animation then shows how the theorem applies to the infinite group which was used as the original example. Finally, the student is introduced to yet another example of a group action - namely conjugation - but this is merely an introduction, the remainder of the work being left to the student after the programme. |
| Master spool number: | HOU3098 |
| Production number: | FOUM009W |
| Videofinder number: | 846 |
| Available to public: | no |
