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Description
This programme investigates the idea of 'isomorphism' in group theory, and shows how two groups can really be the same, even though they may look totally different at first sight.
This programme investigates the idea of 'isomorphism' in group theory, and shows how two groups can really be the same, even though they may look totally different at first sight.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 04C |
| First transmission date: | 1991 |
| Published: | 1991 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:15 |
| + Show more... | |
| Producer: | Andrew Adamyk |
| Contributors: | David Brennan; Ian Harrison |
| Publisher: | BBC Open University |
| Keyword(s): | Arthur Cayley; Isomorphism class; Modulo addition; Modulo addition/multiplication; Scottish country dancing |
| Subject terms: | Graph theory; Isomorphisms (Mathematics); Mathematics; Representations of groups |
| Footage description: | The prograrrme begins with a Scottish Country Dance for four couples, and poses the Question - what's the connection between this dance and addition of integers. David Brannan shows how you can consider the dance as the composition of four particular permutations (or moves) of the couples involved. Ian Harrison shows how these moves can be combined to form a group, and records the group using a Cayley Table. David Brannan shows how modulo addition of the first four integers leads to another group. Ian demonstrates that the Cayley tables for both these groups have the same pattern, despite the groups having different elements. David introduces another group, using modulo multiplication, and shows that its Cayley Table can be rearranged to give the same pattern as the other two groups. Ian introduces two more groups with four elements with Cayley Tables which cannot be made to have the same pattern as the first three. David notes that groups with the same pattern in their Cayley Table are said to be "Isomorphic" groups. Ian shows that there are only two types of Cayley Table pattern for groups with six elements. David now develops a definition of the term "Isomorphic", by showing the conditions for two groups to have the same pattern in their Cayley Tables. Finally he shows how groups can be classified into isomorphism classes. (This programme was re-edited in 1995 as FOUM471L) |
| Master spool number: | HOU6440 |
| Production number: | FOUM343A |
| Videofinder number: | 3636 |
| Available to public: | no |
