Available online
Description
This programme examines the characteristics of permutations and cycles. Particular attention is paid to the features of the conjugacy classes of permutations.
This programme examines the characteristics of permutations and cycles. Particular attention is paid to the features of the conjugacy classes of permutations.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 14 |
| First transmission date: | 22-05-1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:28 |
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| Producer: | Jack Koumi |
| Contributors: | Alan Best; Roger Duke |
| Publisher: | BBC Open University |
| Keyword(s): | Animation; Symmetric |
| Footage description: | Roger Duke introduces the programme. He defines a permutation and then discusses the notation conventions for permutations. The notation gives a complete cycle which tells us exactly what the permutation does to each element. Using animations Roger gives several examples of different types of cycles. He shows that any permutation can be expressed as a combination of disjoint cycles. Alan Best argues that the real interest in permutations lies in the fact that all the permutations of a given set form a group. They are also functions. Permutations are therefore multiple by the composition rule. He shows how this is done by multiplying a 5 cycle by a 3 and 2 cycle. Roger Duke now looks at the group of permutations for the group S3. This consists of a one cycle three transpositions or two cycles and two three cycles. He then shows how the permutations of the cycle correspond to the symmetries of the equilateral triangle. The transpositions correspond to reflections and the three cycles correspond to rotations. He then discusses the relation between the eye structure and the conjugacy classes for the symmetry group of the triangle. Alan Best recaps on the definition of conjugacy and after discussing some further points of notation convention, works through the conjugation formula to show how two cycles can be shown to be conjugate. He argues from this result that the conjugacy class of the transposition (12) is the set of all transpositions. In order to show this he works through some more examples with the permutations written out in 2 line form. He then works through the conjugacy formula to show that the conjugate K(l2)K⁻¹ is the transposition K(l) K(2). He then shows that any conjugate of a three cycle is a three cycle. From these results he derives the rule that any permutation is conjugate to all other permutations with the same cycle form and that any two permutations having the same cycle structure must be conjugate. Roger Duke reviews the findings of the programme demonstrates their importance by relating them to Cayley's Theorem that every finite group is isomorphic to a group of permutations. |
| Master spool number: | HOU3011 |
| Production number: | FOUM015K |
| Videofinder number: | 842 |
| Available to public: | no |
