Here are (some of) the topics that will be covered in this class.
- 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.
- 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.
- Cyclic groups and classification of their subgroups (Fundamental Theorem of cyclic groups). Order of an element.
- Permutation groups. Cyclic notation of permutations. Transpositions. Order and sign of permutations. Alternating groups.
- Isomorphisms of groups. Cayley's Theorem. Properties of isomorphisms. (Inner) automorphisms.
- Cosets. Lagrange's Theorem and applications. Index of a subgroup. Orbit-stabiliser Theorem.
- External direct products of groups and their properties. Applications.
- Normal subgroups and normal subgroup test. Factor groups and internal direct products.
- Group homomorphisms and properties. First Isomorphism Theorem.
- Fundamental Theorem of Finite Abelian groups. Isomorphism classes of abelian groups.
If there is time, we will cover additional topics as selected by the class.
- Definition of a ring and subrings. Basic properties.
- Integral domains and Fields. Characteristic of a ring.
- Ideals and factor rings. Prime ideals and maximal ideals.
- Ring homomorphisms and properties. Field of quotients.
- Polynomial rings and the division algorithm. Polynomial factorisation and irreducibility. Principal ideal domains.
- Divisibility in integral domains. Unique factorisation domains, Euclidean domains.