Fundamentals of Pure Mathematics 2012/13 Course Outline

1. Graphs, symmetries of a graph, examples.
2. Operations, definition of a group, symmetries of a graph form a group, examples of groups.
3. Dihedral groups [JJ, p54] and symmetric groups [JJ, 2].
4. Rotations of Platonic solids, symmetries of vector spaces, first consequences of the axioms [JJ, 4.5], commutativity [JJ, p41].
5. Product groups [JJ, 4.6], subgroups [JJ, 5], test for a subgroup [JJ, 5.1], examples [JJ, 5.2].
6. Order of elements [JJ, 6.3], cyclic groups [JJ, 6.1] and subgroups [JJ, 6.2].
7. Behaviour of cyclic groups under products [JJ, 6.7] and subgroups [JJ, 6.6], recap on equivalence relations [JJ, 8].
8. Rules for cosets, proof of Lagrange [JJ, 10.1-10.3].
9. Consequences of Lagrange [JJ, 10.4].
10. Homomorphisms and Isomorphisms [JJ, 9.1], examples, basic properties [JJ, 9.2], kernels and images [JJ, 15.1].
11. Products and Isomorphisms [JJ, 14.3 p145] and examples. Definition of a group action [JJ, 7] and examples.
12. Faithful actions [JJ, 15.2], Group actions=homomorphisms [JJ, 7.4, 9.3], Cayley's Theorem [JJ, 9.4].
13. Cayley's Theorem examples and remarks, Orbits and Stabilizers [JJ, 7.2, 7.3] and examples, transitive actions [JJ, p83].
14. Statement and proof of the Orbit--Stabilizer Theorem [JJ, p117], practical applications to Platonic solids, Cauchy's Theorem [JJ, p144].
15. Polya Counting [JJ, 11.3] and examples [JJ, 12].
16. Symmetric and Alternating Groups (mainly revision [JJ, 2], [Liebeck, 20]), with applications to Platonic Solids.
17. Symmetries of tetrahedron and dodecahedron, then conjugacy classes [JJ, 13.1, 13.2], centralizers [JJ, 13.4], centres of groups [JJ, 13.5], the class equation [JJ, p135].
18. Two applications of the class equation [JJ, 13.6, Thm 5] and [JJ, Thm 11 p173], conjugacy in S_n [JJ 13.3], normal subgroups [JJ, 15.4] and their relationship with conjugacy classes.
19. Examples of normal subgroups: centres [JJ, Ex 5 p155], kernels of actions [JJ, 15.2], kernels of group homomorphisms [JJ, 15.1] and [JJ, Thm 6 p155]. Index two subgroups are normal [JJ, Thm 8, p156] and applications.
20. Overview of factor groups and why we care [JJ, Chapter 16], then overview of the course.