Here is a basic outline of the topics we will cover (time permitting!) this semester.
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.
By the end of the course, you should
- be able to define a group and give some simple examples of such objects.
- be able to prove simple results about groups.
- understand important concepts in group theory, including
- Cyclic groups
- Permutation Groups
- Group products
- Normal subgroups
- Homomorphisms and the First Isomorphism Theorem.
- be able to cope with abstract mathematical ideas and observe how they allow the proof of general results.