# GROUPS & RINGS - 2022/3

Module code: MAT2048

## Module Overview

This module provides an introduction to abstract algebra through elementary group and ring theory.  Thus it forms the starting point for all the algebraic modules that follow: MAT2005 Algebra and Codes, MAT3011 Galois Theory, MAT3032 Advanced Algebra and MATM011 Lie Algebras.

### Module provider

Mathematics

PRINSLOO Andrea (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 95

Seminar Hours: 11

Guided Learning: 11

Captured Content: 33

Semester 1

MAT1031 Algebra

## Module content

Indicative content includes:

• Revision of the group axioms, permutations and the integers modulo n.

• Symmetries and the dihedral groups.

• Cyclic groups, direct products.

• Group homomorphisms and isomorphisms.

• Cosets, normal subgroups, quotient groups. Lagrange's theorem.

• Introduction to rings and fields.  Subrings, ideals and quotient rings.

• Rings of polynomials. Construction of finite fields.

## Assessment pattern

Assessment type Unit of assessment Weighting
Online Scheduled Summative Class Test ONLINE TEST 20
Examination Online ONLINE EXAM 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate their ability to

·         construct and interpret mathematical arguments in the context of this module;

·         display subject knowledge by recalling key definitions and results;

·         apply the techniques learnt to both routine and unfamiliar problems.

Thus, the summative assessment for this module consists of:

·         One two-hour examination at the end of Semester 1, worth 80% of the module mark.

·         In-semester test, worth 20% of the module mark.

Formative assessment and feedback

Students receive written comments on their marked coursework assignments. Maple TA may be used in assignments to provide immediate grading and feedback. Verbal feedback is provided in lectures and office hours.

## Module aims

• introduce the axiomatic approach to group theory and ring theory
• develop confidence in working with algebraic structures
• provide a firm foundation for subsequent study of abstract algebra

## Learning outcomes

 Attributes Developed 1 Know the definition of a groupt and recognise  and recognize standard examples K 2 Carry out calculations involving groups, selecting an appropriate method. KC 3 Have insight into the structure of groups, their subgroups and mappings between groups. KC 4 Know the definition of a ringt and recognise  and recognize standard examples. K 5 Understand rings of polynomials and the use of quotient rings in constructing finite fields. KC 6 Construct simple proofs similar to those encountered in the module. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• An introduction to algebraic structure theory

• Experience of the methods used to interpret, understand and solve problems in group and ring theory

The learning and teaching methods include:

• Three 50-minute lectures per week for eleven weeks, some being used as tutorials, problem classes and in-semester tests.

• Online notes supplemented by additional examples in lectures.

• Two unassessed coursework assignments, marked and returned.

• Personal assistance given to individuals and small groups in office hours.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT2048

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics and Physics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2022/3 academic year.