MTH5141 - Computational group theory - 2019

Groups are abstract mathematical objects capturing the concept of symmetry, and therefore are ubiquitous in many mathematical disciplines and other fields of science, such as physics, chemistry, and computer science. This unit is an introductory course on group theory and computational methods, using the computer algebra system GAP (www.gap-system.org). This unit will cover a selection of topics from the following list. Abstract Groups: knowing the basic definitions and standard results; Group Actions: orbits, stabilisers, and the orbit-stabiliser theorem; Group Presentations: free groups, abelian invariants, Todd-Coxeter algorithm; Permutation Groups: stabiliser chains, bases and strong generating sets, membership test; Nilpotency and Solvability: knowing the basic definitions and properties. Polycyclic Groups: polycyclic series and generating sets, polycyclic presentations; GAP: learn how to use the computer algebra system GAP to compute with groups. Some of the material will be self-taught through guided reading.

On completion of this unit students will be able to:

1. Formulate complex problems using appropriate terminology in algebra
2. Demonstrate a profound understanding of abstract concepts in group theory
3. Appreciate the nature of algebraic proofs, be able to use a variety of proof-techniques unique to working with groups;
4. Apply a variety of expert algorithms for different algebraic objects, in particular, groups
5. Use the computer algebra system GAP to compute with groups and related structures.

Examination (3 hours): 60% (Hurdle)

Continuous assessment: 40%

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for the end-of-semester exam.

This unit is offered at both Level 4 and Level 5, differentiated by the level of the assessment. Students enrolled in MTH5141 will be expected to demonstrate a higher level of learning in this subject than those enrolled in MTH4141. The assignments and exam in this unit will use some common items from the MTH4141 assessment tasks, in combination with several higher level questions and tasks.

Master of Mathematics