# Module Catalogue

Throughout the 2020-21 academic year, we will be adapting the way we teach and assess your modules in line with government guidance on social distancing and other protective measures in response to Coronavirus. Teaching will vary between online and on-campus delivery through the year, and you should read guidance from the academic department for details of how this will work for a particular module. You can find out more about the University’s overall response to Coronavirus at: https://warwick.ac.uk/coronavirus.

# MA136-6 Introduction to Abstract Algebra

Department
Warwick Mathematics Institute
Level
David Wood
Credit value
6
Module duration
5 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

##### Introductory description

This module introduces important algebraic structures including groups, rings and fields. Students will learn how to verify that a set is a group, ring or field, and how to carry out elementary operations in these structures. They will understand the relation between a group, a subgroup and the cosets of a subgroup which leads to Lagrange's theorem. They will also assimilate permutations, symmetric groups, and alternating groups, and know how to determine the unit group of a ring.

##### Module aims

To introduce First Year Mathematics students to abstract Algebra, covering Group Theory and Ring Theory

##### Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Section 1 Group Theory
Motivating examples: numbers, symmetry groups
Definitions, elementary properties
Subgroups, including subgroups of Z
Arithmetic modulo n and the group Z_n
Lagrange's Theorem
Permutation groups, odd and even permutations (proof optional) Normal subgroups and quotient groups

Section 2 Ring Theory
Definitions: Commutative and non-commutative rings, integral domains, fields
Examples: Z, Q, R, C, Z_n, matrices, polynomials, Gaussian integers
Unit, unit groups, factorisation, examples where unique factorisation fails

##### Learning outcomes

By the end of the module, students should be able to:

• Understand the abstract definition of a group, and be familiar with the basic types of examples, including numbers, symmetry groups and groups of permutations and matrices.
• Understand what subgroups are, and be familiar with the proof of Lagrange’s Theorem.
• Understand the definition of various types of ring, and be familiar with a number of examples, including numbers, polynomials, and matrices.
• Understand unit groups of rings, and be able to calculate the unit groups of the integers modulo n.

Any library book with Abstract Algebra in the title would be useful.

##### Subject specific skills

This module introduces important algebraic structures including groups, rings and fields. Students will learn how to verify that a set is a group, ring or field, and how to carry out elementary operations in these structures. They will understand the relation between a group, a subgroup and the cosets of a subgroup which leads to Lagrange's theorem. They will also assimilate permutations, symmetric groups, and alternating groups, and know how to determine the unit group of a ring.

##### Transferable skills

The module reinforces logical thinking and deductive reasoning which are valuable transferable skills. The algebraic structures introduced are the heart of modern cryptography and information security.

## Study time

Type Required
Lectures 15 sessions of 1 hour (25%)
Tutorials 5 sessions of 1 hour (8%)
Private study 40 hours (67%)
Total 60 hours
##### Private study description

40 hours private study, revision for exams, and non-assessed assignments

## Costs

No further costs have been identified for this module.

You do not need to pass all assessment components to pass the module.

##### Assessment group D1
Weighting Study time
Weekly assignments 15%
2 hour online examination (Summer) 85%

Exam

• Online examination: No Answerbook required
##### Assessment group R
Weighting Study time
exam 100%
##### Feedback on assessment

Marked assignments and exam feedback.

## Courses

This module is Core for:

• Year 1 of UMAA-G100 Undergraduate Mathematics (BSc)
• Year 1 of UMAA-G103 Undergraduate Mathematics (MMath)
• Year 1 of UMAA-G106 Undergraduate Mathematics (MMath) with Study in Europe
• Year 1 of UMAA-G1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
• Year 1 of UMAA-GL11 Undergraduate Mathematics and Economics
• Year 1 of UECA-GL12 Undergraduate Mathematics and Economics (with Intercalated Year)
• Year 1 of UMAA-GV17 Undergraduate Mathematics and Philosophy
• Year 1 of UMAA-GV18 Undergraduate Mathematics and Philosophy with Intercalated Year
• Year 1 of UMAA-G101 Undergraduate Mathematics with Intercalated Year

This module is Optional for:

• Year 1 of UPXA-FG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
• Year 1 of UPXA-GF13 Undergraduate Mathematics and Physics (BSc)
• Year 1 of UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)
• Year 1 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
• Year 1 of USTA-GG14 Undergraduate Mathematics and Statistics (BSc)

This module is Option list A for:

• Year 1 of UCSA-G4G1 Undergraduate Discrete Mathematics
• Year 1 of UCSA-G4G3 Undergraduate Discrete Mathematics