BMTS209 - IfE

BMTS209
Group Theory

MQF Level: 6

ECTS Value: 5 ECTS

Self-Study Hours: 60

Duration: 10 Sessions

Contact Hours: 25

Mode of Delivery: Blended

Assessment Hours: 40

Entry Requirements

Applicants applying for the module are to be in possession of the following:

1 An MQF Level 3 (minimum Grade 5 or C) in Maltese, English Language and Mathematics awarded by MATSEC or an equivalent examination body recognised by the IfE

AND

A minimum of one of the following:
a) An awarded MATSEC Certificate or equivalent (MQF Level 4) with a Grade C or better in Mathematics at Advanced Level; OR
b) Three subjects at advanced level (MQF Level 4) including a Grade C or better in Mathematics and another subject, and at least a Grade D in a third subject; OR
c) Two subjects at Advanced Level (MQF Level 4) at Grade C or better including Mathematics, and three intermediate subjects with a minimum Grade D.

Overall Objectives and Outcomes

The unit introduces the course participants to basic terminology related to Group Theory that will lead to the definition of what a Group is. This will enable the participants to not only understand different examples of Groups but to come up themselves with other examples. This unit shall then focus on the distinction between homomorphisms and isomorphisms, as well as examples of each. Two important theorems that will be proven are Lagrange’s Theorem and Cayley’s Theorem.

Throughout the unit, emphasise will not only be on the definitions and theorems to which they lead but on examples that will give a better understanding of these same definitions and theorems.

By the end of this module, the learner will be able to:

Competences

Illustrate the meaning and difference between onto and one-to-one mappings by using a diagram;
Differentiate between equivalence and equality and illustrate their use in mathematical reasoning.
Verify if a mapping is a homomorphism or an isomorphism;
Prove that a set is a group by proving the four axioms of Groups;
Develop curriculum related material to be used to explain basic concepts of Group Theory;
Apply definitions and theorems to decide if a set is a group or not, and if yes what special properties it has.

Knowledge

Recall basic definitions of Set Theory;
Differentiate between onto and one-to-one mappings and identify these properties in given functions;
Distinguish between Abelian and non-Abelian Groups;
Explain the Equivalence Relation Lemma;
Distinguish between homomorphisms and isomorphisms;
Explain the difference between right and left cosets;
State Cayley’s Theorem;
Define the meaning of Symmetric Groups;
Define the meaning of Conjugacy.

Skills

Give simple examples of Groups demonstrating a clear understanding of the fundamental properties that define group structures;
Provide and analyse examples of Symmetric Groups, explaining their significance in the context of permutations and group actions and demonstrating how these groups relate to the broader theory of algebraic structures;
Show that conjugacy is an equivalence relation and apply this concept to analyse the structure of groups.
Apply Cayley’s Theorem to demonstrate that every group is isomorphic to a subgroup of a symmetric group.

Assessment Methods

This module will be assessed through: Class Contribution, Presentation.