Available online
Description
This programme explains the arguments behind Lagrange's Theorem and then examines its significance as a means of analysing groups.
This programme explains the arguments behind Lagrange's Theorem and then examines its significance as a means of analysing groups.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 06 |
| First transmission date: | 20-03-1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:27 |
| + Show more... | |
| Producer: | James Burge |
| Contributors: | Alan Best; Roger Duke |
| Publisher: | BBC Open University |
| Keyword(s): | Group theory; Hexagon; Prime numbers; Symmetry |
| Footage description: | Roger Duke states that there is only one group of order 1979 Lagrange's theorem explains why this is so. Alan Best discusses Lagrange's theorem. It is based on the idea of cosets. He defines a coset and then gives some detailed examples, by forming three cosets of the group of all symmetries of the regular hexagon. The three cosets use up all twelve elements of the group. There are twelve possible cosets of this group, but the other cosets are shown to be repetitions of the ones already formed. Alan Best argues from this result that cosets partition a finite group into subsets of equal numbers of elements. This is the basis of Lagrange's theorem. Roger Duke now examines a different subgroup of the symmetry group of the hexagon. This subgroup consists of three rotations. He then partitions the twelve members of the group into four cosets each of three elements. This confirms the result shown by Alan. Alan Best now explains the significance of these results. To get the partitioning effect into cosets, the order of the subgroup must divide the order of the group. This is Lagrange's theorem. The theorem tells us which subgroups a symmetry group might have and which it cannot have. Roger Duke now returns to group order 1979. As it is a prime number. Lagrange's theorem states that it cannot have any subgroups except for 1 and the whole group. Alan Best takes another prime number 5 and shows that there is only one group of order 5 or for any prime number, such as 1979. |
| Master spool number: | HOU2935 |
| Production number: | FOU9022K |
| Videofinder number: | 838 |
| Available to public: | no |
