Aims

To introduce students to the basic ideas of group theory.

To equip students with a range of basic tools and methods for analysing geometric and algebraic structures. To enable the students to apply these techniques to naturally occurring phenomena involving symmetries or transformations. To reinforce the students’ ability to read, understand and develop mathematical proofs.

Module Summary

The module introduces the axioms of a group and explores many examples, particularly of symmetry groups,permutation groups, and matrix groups. Basic properties of groups will be demonstrated, always with reference to examples.

We shall prove Lagrange's Theorem, which tells us that for finite groups the number of elements in a subgroup divides the number of elements in the parent group. Groups of permutations will be studied systematically.
We will introduce group homomorphisms and prove the Isomorphism Theorem which associates an
isomorphism to each homomorphism. We also consider group actions, and Cayley's theorem and apply group
actions to prove Cauchy’s and Sylow’s theorems, which are partial converses to Lagrange’s.

Groups arise naturally as concise and tractable characterisations of geometries: for example as symmetries of regular Euclidean figures, of lattices and of graphs and their higher dimensional analogues. The interaction between group theory and geometry will be the main focus of this course. Various examples of groups given by presentations and groups acting on spaces, such as polyhedra or graphs will be studied, with the emphasis on groups acting on graphs.

Outline Of Syllabus

To introduce students to the basic ideas of group theory.

Module Summary
The course introduces the axioms of a group and explores many examples, particularly of symmetry groups,permutation groups, and matrix groups. Basic properties of groups will be demonstrated, always with reference to examples.
We shall prove Lagrange's Theorem, which tells us that for finite groups the number of elements in a subgroup divides the number of elements in the parent group. Groups of permutations will be studied systematically.
We will introduce group homomorphisms and prove the Isomorphism Theorem which associates an
isomorphism to each homomorphism. We also consider group actions, and Cayley's theorem and apply group
actions to prove Cauchy’s and Sylow’s theorems, which are partial converses to Lagrange’s.

Graph theory.
Symmetries of regular polygons and of graphs.
Group actions on graphs and Cayley graphs.
Free groups and Stallings foldings. Presentations of groups.

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.
Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.