Available online
Description
This programme introduces the notion of isomorphism, and looks at isomorphisms between groups of order up to 6.
This programme introduces the notion of isomorphism, and looks at isomorphisms between groups of order up to 6.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 04 |
| Recording date: | 02-06-1978 |
| First transmission date: | 06-03-1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:25:00 |
| + Show more... | |
| Producer: | James Burge |
| Contributors: | Roger Duke; Chris Rowley |
| Publisher: | BBC Open University |
| Keyword(s): | Cayley table; Element; Group; Isomorphism |
| Subject terms: | Graph theory; Isomorphisms (Mathematics); Mathematics; Representations of groups |
| Footage description: | Roger Duke shows that the symmetry group of an equilateral triangle has six elements. From these elements he makes a composition table which shows all possible combinations of elements. He then constructs a composition group for the isometrics of the plane that carry three points to themselves. This table is identical to the first one, a correspondence exists between the elements in the composition tables, this is called an isomorphism. The correspondence between two groups is not always obvious. Roger Duke looks at a figure with just two symmetries; an identity and a rotation. Chris Rowley looks at another figure with two symmetries; an identity and a reflection. He shows that the two correspondence tables for the groups are the same. There can only be one to one correspondence between two groups if the groups have the same number of elements, the same order. Roger Duke looks now at a group of order three, a symmetry group for a figure the elements of which are identity, a rotation through one third of a revolution and two thirds of a revolution. He compiles the composition table. Chris Rowley compiles another composition table for another group of order 3 the integers modulo 3 under addition. The two groups are now compared to see if they are isomorphic. A correspondence between the elements is found. The groups are isomorphic. Roger and Chris now try to find a method for testing if much larger groups are isomorphic. They discover that a condition for an isomorphism is that, for a certain correspondence, compositions in one group correspond to compositions in the other. Roger Duke looks at the symmetry of a figure, giving a group of order four. He compiles a composition table for this group. Chris Rowley goes through the same process for a three-dimensional figure of symmetry group order four. They then compare the two tables, which are shown not to be isomorphic, as corresponding elements do not compose to give corresponding elements. Roger Duke now considers groups of order six. He looks at the symmetry of a figure giving a group of order six. He compares this with the table for the group of order six drawn at the beginning of the programme, and shows that the two are not isomorphic. Finally Roger Duke looks at the symmetry group of an infinite saw blade. This group has an infinite number of elements. He then shows that although it is impossible to draw the composite table for this group, it is isomorphic to the group of integers under addition, by using the condition obtained previously that compositions in one group correspond to compositions in the other. He then sums up the significance of the results, obtained in the programme which allow whole classes of groups to be considered at one time. This is the basis of abstract group theory, one of the main tools of modern mathematics. |
| Master spool number: | HOU2900 |
| Production number: | 00525_4284 |
| Videofinder number: | 836 |
| Available to public: | no |
