Available online
Description
This programme looks at the properties of the group of direct symmetries of the dodecahedron. These properties are then compared to A5, the group of even permutations of five symbols to show that t...hey are isomorphic. The geometric properties of the dodecahedron are then used to show that A5 is a simple group.
This programme looks at the properties of the group of direct symmetries of the dodecahedron. These properties are then compared to A5, the group of even permutations of five symbols to show that t...hey are isomorphic. The geometric properties of the dodecahedron are then used to show that A5 is a simple group.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 15 |
| First transmission date: | 17-11-1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:29 |
| + Show more... | |
| Producer: | Jack Koumi |
| Contributors: | Colin Rourke; Ian Stewart |
| Publisher: | BBC Open University |
| Keyword(s): | Cubes; Models; Symmetry |
| Footage description: | Colin Rourke introduces the programme which is about the group of direct symmetries of the dodecahedron and its relationship to the group of even permutations of five symbols, A5. Ian Stewart describes the geometry of the dodecahedron. It has twelve faces, twenty vertices and thirty edges. The group of rigid body motions consists of sixty elements, sixty direct symmetries. Using models of the dodecahedron Ian and Colin describe the possible angles of rotational symmetry which make up the elements of the group. Ian Stewart now compares this classification to the group A5. If the two groups are isomorphic then the different elements of the groups must match. To show that the groups are isomorphic he argues that he must demonstrate that each rotational symmetry permutates five things and that the cycle types match those of A5. Using a series of models of cubes placed within the dodecahedron Colin Rourke shows that it is possible to locate 5 cubes inside this space the edges of which also form a diagonal of one of the faces of the dodecahedron. He then shows how rotations of the dodecahedron permutate the relative positions of the cubes. Each vertex rotation generates a three cycle movement of the cubes. Edge axis rotations permute four cubes in pairs, corresponding to disjoint two-cycles. Face axis rotations give a five cycle. These match the cycle groups of A5. Each rotation can be represented by whichever permutation it produces on the cubes. In order to show that A5 and the dodecahedron group are isomorphic Colin Rourke demonstrates that their relationship is one-one and preserves combinations in the two groups. Ian Stewart argues that this isomorphism allows us to study the group A 5 to show that it is a simple group, a group with no normal sub-groups. Normal sub-groups have to be unions conjugacy classes. The conjugacy classes of A5 can be seen by looking at the dodecahedron. Colin Rourke defines conjugacy and explains its significance in geometric terms. He shows that the edge axis rotations all lie in the same conjugacy class. Next Colin and Ian show that all the vertex rotations lie in a single conjugacy class, and similarly they show that the 24 face-axis rotations split into two conjugacy classes. Ian Stewart relates these results to A5. As the the groups are isomorphic then conjugacy classes are the same. He now looks for any normal sub-groups in A5, unions of conjugacy classes which divide the order of A5, 60. There is no union which satisfies this requirement. Therefore A5 is simple. |
| Master spool number: | HOU3065 |
| Production number: | FOUM016E |
| Videofinder number: | 843 |
| Available to public: | no |
