Syllabus

Here are (some of) the topics that will be covered in this class.

Groups

1. Definition of a group and understanding of basic properties of groups, as well as some fundamental examples of groups arising in geometry, linear algebra etc.
2. Definition of a subgroup. Subgroup tests and how to use them in important contexts. Finite groups and the order of groups. Generators. Centres and Centralisers.
3. Cyclic groups and classification of their subgroups (Fundamental Theorem of cyclic groups). Order of an element.
4. Permutation groups. Cyclic notation of permutations. Transpositions. Order and sign of permutations. Alternating groups.
5. Isomorphisms of groups. Cayley's Theorem. Properties of isomorphisms. (Inner) automorphisms.
6. Cosets. Lagrange's Theorem and applications. Index of a subgroup. Orbit-stabiliser Theorem.
7. External direct products of groups and their properties. Applications.
8. Normal subgroups and normal subgroup test. Factor groups and internal direct products.
9. Group homomorphisms and properties. First Isomorphism Theorem.
10. Fundamental Theorem of Finite Abelian groups. Isomorphism classes of abelian groups.

Rings

1. Definition of a ring and subrings. Basic properties.
2. Integral domains and Fields. Characteristic of a ring.
3. Ideals and factor rings. Prime ideals and maximal ideals.
4. Ring homomorphisms and properties. Field of quotients.
5. Polynomial rings and the division algorithm. Polynomial factorisation and irreducibility. Principal ideal domains.
6. Divisibility in integral domains. Unique factorisation domains, Euclidean domains.