Learning Outcomes:
On completion of this module, the student is expected to:- demonstrate an understanding of the theoretical aspects covered in this module;- be able to carry out any computations involving the objects described in this module (such as finding the orbits of a group action, finding the Smith normal form of a matrix, decomposing a module into invariant factors, reducing a matrix to Jordan canonical form, for example).
Indicative Module Content:
Review of the first, second and third isomorphism theorems for groups and rings. Subgroups generated by a subset, generators and relations. Group actions, orbit-stabilizer theorem, Burnside’s lemma. Sylow theorems, other examples and applications. Linear groups.
Ideals and subrings generated by subsets, review of principal ideal domains, unique factorization domains, Euclidean domains. The Chinese Remainder Theorem. Matrices over EDs and PIDs, the Smith normal form. Modules, free modules and torsion modules. Finitely generated modules over a PID, and the structure theorem. Applications including the classification of finite abelian groups, Jordan canonical form, rational canonical form.