MTH4141 - Computational group theory

There is a more recent version of this academic item available.

Overview

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. 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.

Offerings

S1-01-CLAYTON-ON-CAMPUS

Rules

Enrolment Rule

Contacts

Chief Examiner(s)

Associate Professor Heiko Dietrich

Unit Coordinator(s)

Associate Professor Heiko Dietrich

Notes

This unit is offered in alternate years commencing Semester 1, 2019.

Learning outcomes

On successful completion of this unit, you should be able to:

Formulate complex problems using appropriate terminology in algebra;

Demonstrate a profound understanding of abstract concepts in group theory;

Appreciate the nature of algebraic proofs, be able to use a variety of proof-techniques unique to working with groups;

Apply a variety of expert algorithms for different algebraic objects, in particular, groups;

Use the computer algebra system GAP to compute with groups and related structures.

Teaching approach

Active learning

The teaching approach for this unit is a blend of lectures and applied classes, involving active learning. The content of the unit will be delivered in 3 hours of lectures; this is an established way to teach high-level abstract mathematics effectively. In this model, you are expected to revise the theory explained in the lectures at home. This active learning process will assist students in fully understanding the new concepts. The written assignments (solving selected problems in computational group theory) have the same effect: students are required to study the lectures in order to derive solutions to the exercises. The theory will be exemplified in the applied classes: this is the place where the theoretical concepts of the lectures are discussed interactively with explicit examples. These classes will also involve group work where you have to work on problems in teams. The computational aspects of the unit will be illustrated with the computer algebra system GAP.

Assessment summary

Examination (3 hours and 10 minutes): 60% (Hurdle)
Continuous assessment: 40%

Hurdle requirement: If you would otherwise have passed the unit but who do not achieve at least 45% of the marks available for the end-of-semester examination will receive a Hurdle Fail (NH) grade and a mark of 45 on your transcript.

Assessment

1 - Assignments

2 - Mid-semester test

3 - Examination (3 hours and 10 minutes)

Scheduled and non-scheduled teaching activities

Applied sessions

Lectures

Workload requirements

Workload

Learning resources

Technology resources

Other unit costs

Costs are indicative and subject to change.
Miscellaneous items required (Printing, Stationery) - $100.

Availability in areas of study

Master of Mathematics