MA1366 Introduction to Abstract Algebra
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 noncommutative 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.
Indicative reading list
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 nonassessed 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

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 UMAAG100 Undergraduate Mathematics (BSc)
 Year 1 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 1 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 1 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 1 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 1 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 1 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)
 Year 1 of UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 1 of UMAAGV18 Undergraduate Mathematics and Philosophy with Intercalated Year
 Year 1 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Optional for:
 Year 1 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 1 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 1 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 1 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 1 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)
This module is Option list A for:
 Year 1 of UCSAG4G1 Undergraduate Discrete Mathematics
 Year 1 of UCSAG4G3 Undergraduate Discrete Mathematics