Available online
Description
This programme investigates the isomorphism theorem in group theory and how this theorem uses homomorphisms to identify quotient groups.
This programme investigates the isomorphism theorem in group theory and how this theorem uses homomorphisms to identify quotient groups.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 04D; 1995 |
| First transmission date: | 1995 |
| Published: | 1995 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:09 |
| Note: | Open University course M203 Introduction to pure mathematics.Open University course M203 Introduction to pure mathematics. |
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| Producer: | Andrew Adamyk |
| Contributors: | David Brannan; Pip Surgey |
| Publisher: | BBC Open University |
| Keyword(s): | Groups; Homomorphism; Isomorphism; Symmetry |
| Subject terms: | Categories (Mathematics); Functional analysis; Functions; Homomorphisms (Mathematics); Isomorphisms (Mathematics) |
| Footage description: | The programme begins with David Brannan (OU) asking two questions about identifying quotient groups; one involves a complex number group, and one involves the symmetry group of a tetrahedron. He then revises what a quotient group is, using the example of the symmetry group of a square. Pip Surgey (BBC) shows how constructing a quotient group can be thought of as a mapping; in fact, a special mapping called a homomorphism. David Brannan introduces the ideas of image and kernel for a homomorphic mapping, and notes that the kernel of any homomorphism Ts a normal subgroup. But it is also true that every normal subgroup is the kernel of some homomorphism, and Pip Surgey uses this fact to state the Isomorphism Theorem. This is then used to identify the two quotient groups presented at the start of the programme. David Brannan identifies the complex number quotient group, and shows how it can be represented pictorially in the complex plane. Pip Surgey revises the symmetries of a tetrahedron, and then David identifies the tetrahedron quotient group. Finally, Pip Surgey shows how this quotient group can be constructed geometrically. |
| Master spool number: | DOU8021 |
| Production number: | FOUM482X |
| Videofinder number: | 3780 |
| Available to public: | no |
