# GROUPS & RINGS - 2025/6

Module code: MAT2048

## Module Overview

This module provides an introduction to abstract algebra, focusing on the theory of algebraic structures called groups and rings. The module builds on preliminary material on groups introduced in MAT1031 Algebra.

This module forms the starting point for subsequent algebraic modules, such as MAT3032 Advanced Algebra, MATM035 Representation Theory and MATM011 Lie Algebras.

### Module provider

Mathematics & Physics

PRINSLOO Andrea (Maths & Phys)

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

Independent Learning Hours: 54

Lecture Hours: 33

Tutorial Hours: 5

Guided Learning: 25

Captured Content: 33

Semester 2

None.

## Module content

Indicative content includes:

• Definitions and examples of groups, rings and fields.

• Symmetric groups and dihedral groups.

• Subgroups and the subgroup test.

• Lagrange’s theorem.

• Cyclic groups and direct product groups.

• Group homomorphisms and group isomorphisms.

• Normal subgroups, cosets and quotient groups.

• Subrings and subfields, and the subring test and subfield test.

• Ring homomorphisms and ring isomorphisms.

• Ideals and quotient rings.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 20
Examination End-of-Semester Examination (2 hours) 80

N/A

## Assessment Strategy

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

• Understanding of subject knowledge, and recall of key definitions and results in the theory of groups and rings.

• The ability to construct simple proofs similar to those in the module.

• The ability to identify and use the appropriate methods to solve problems involving groups and rings.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1, 2, 4 and 6.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 6.

Formative assessment

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback

Students will receive immediate automated feedback on the first formative unassessed coursework which will run in Mobius software. Students will receive individual written feedback on the second formative unassessed coursework. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in tutorials and office hours.

## Module aims

• Introduce students to basic definitions and results in the theory of groups and rings.
• Develop students¿ understanding of rigorous proofs in the context of abstract algebra.
• Provide students with a firm foundation in abstract algebra underlying subsequent algebraic modules.

## Learning outcomes

 Attributes Developed 001 Students will know the definitions of a group, ring and field, and be able to provide and recognise standard examples of these algebraic structures. KC 002 Students will solve problems involving groups and subgroups, and mappings between groups. KC 003 Students will solve problems involving rings and subrings, and mappings between rings. KC 004 Students will be able to apply the subgroup test, subring test and subfield test. KC 005 Students will be able to construct quotient groups and quotient rings. KC 006 Students will be able to 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 students with an introduction to the theory of groups and rings, and experience understanding and constructing proofs of standard results in this area of abstract algebra.

• Provide students with experience of methods used to interpret, understand and solve problems in the theory of groups and rings.

The learning and teaching methods include:

• Three one-hour lectures for eleven weeks, with typeset module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Five biweekly one-hour tutorials per semester. These tutorials provide an opportunity for students to gain feedback and assistance with end-of-chapter exercises in the module notes, and to practice mock test and mock exam questions under invigilated, closed-book conditions.

• Two unassessed courseworks to provide students with further opportunity to consolidate learning. The first coursework is run in Mobius software to provide students with an opportunity to enhance their digital capabilities and with immediate feedback on answers. The second coursework involves a written submission upon which students receive individual written feedback as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

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

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MAT2048 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing. Students also complete a formative digital assessment run in Mobius software which enables them to further develop their digital capabilities.

Employability: The module MAT2048 equips students with skills which significantly enhance their employability. The mathematical proficiency gained hones critical thinking and problem-solving abilities. Students learn to evaluate complex algebraic problems, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

Global and Cultural Capabilities: Student enrolled in MAT2048 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MAT2048 is a module which demands a rigorous approach to abstract algebra, to which students will learn to adapt. They will gain skills in analysing problems and lateral thinking. Students will complete assessments which challenge them and build resilience.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Optional 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 2025/6 academic year.