Available online
Description
This programme explores what Lagrange's Theorem tells us about subgroups, and introduces the idea of a co-set.
This programme explores what Lagrange's Theorem tells us about subgroups, and introduces the idea of a co-set.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 02D; 1995 |
| First transmission date: | 1994 |
| Published: | 1995 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:07 |
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| Producer: | John Jaworski |
| Contributors: | Ian Harrison; Robin Wilson |
| Publisher: | BBC Open University |
| Keyword(s): | Blocking; LaGrange's Theorem; Order; Subgroups |
| Subject terms: | Algebra, Homological; Categories (Mathematics); Lagrange equations; Set theory |
| Footage description: | Large Cayley Tables do not readily yield their structure - at a glance, they can seem to be just a large confusing array of symbols. But sometimes a hint of a simpler underlying pattern emerges. For example, the 24 x 24 table of the symmetries of a tetrahedron has a simpler 2 x 2 structure if you ignore everything about the individual symmetries except whether they are direct or indirect. Superimposing the 2 x 2 structure on the 24 x 24 table gives a blocking of the Cayley Table. The search for blockings begins with subgroups, and a first question might be 'what subgroups can you have? LaGrange's Theorem gives the answer, that the order of a subgroup must divide the order of the full group. But the proof of LaGrange's Theorem introduces the concept of cosets: they don't appear in the statement of the theorem, but they do appear in the proof. So what are they? Answering that question involves building new (physical) models of group structures, and those models lead us back to blockings. For symmetry groups, each coset corresponds to a different orientation of a colouring of the object. Different colourings give different cosets, and whether the cosets correspond to a blocking or not depends on whether the subgroup that preserves the colouring is normal. This can be detected graphically, by examining the models. And the block structure can be identified as a group in its own right, the quotient group. The programme ends with the construction of a large model of the tetrahedron group, which yields another blocking of the original table - which turns out to be the same group that has been analysed in smaller models throughout the programme. |
| Master spool number: | DOU8019 |
| Production number: | FOUM480J |
| Videofinder number: | 3278 |
| Available to public: | no |
