MA26810 Algebra 3
Introductory description
This is a second abstract algebra module for Mathematics students.
Module aims
It is a second Abstract Algebra module, where the students should get workable knowledge of many algebra concepts. Compare to joint degree students, doing Groups and Rings, the students will get extended knowledge of several topics.
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.
 Group Theory: quaternionic group, matrix group, coset, Lagrange’s theorem, quotient group, isomorphism theorem, free group, group given by generators and relations, group action, Gset G/H, orbit, stabiliser, the orbitstabiliser theorem, conjugacy class, classes in S_n, classification of groups up to order 8.
 Ring Theory: domain, isomorphism theorem, Chinese remainder theorem for Z/nZ and F[x]/(f), unit group, prime and irreducible element, factorization, Euclidean domain, characteristic of a field, unique factorization domain, ED is UFD, finite subgroup of units in fields.
 Module Theory: module, free module, internal and external direct sum, free abelian group, unimodular Smith normal form, the fundamental theorem of finitely generated abelian groups.
 List of covered algebraic definitions: direct product, coset, normal subgroup, quotient group, ideal, quotient ring, domain, irreducible element, prime element, euclidean domain, unique factorisation domain, direct product, free group, generators and relations, module, free module, direct sum, unimodular Smith normal form, action, orbit, stabiliser, fixed points.
Learning outcomes
By the end of the module, students should be able to:
 have a working knowledge of the main constructions and concepts of theories of groups and rings
 recognise, classify and construct examples of groups and rings with specified properties by applying the algebraic concepts
Indicative reading list
Ronald Solomon, Abstract Algebra, Brooks/Cole, 2003.
Niels Lauritzen, Concrete Abstract Algebra, Cambridge University Press, 2003
John B. Fraleigh, A first course in abstract algebra, Pearson, 2002
Joseph A. Gallian, Contemporary Abstract Algebra, Cengage Learning, 2012
View reading list on Talis Aspire
Subject specific skills
Students will improve their skills in thinking algebraically in a variety of settings. This includes working with axiomatic definitions of algebraic objects and analysing the structure and relationships between algebraic objects using fundamental tools such as subobjects and homomorphisms, laying a foundation for future study in algebra, number theory and algebraic geometry.
Transferable skills
The module emphasises the power of generalisation and abstraction. Students will improve their ability to analyse abstract concepts and to solve problems by selecting and applying appropriate abstract tools.
Study time
Type  Required 

Lectures  20 sessions of 1 hour (20%) 
Online learning (independent)  9 sessions of 1 hour (9%) 
Private study  13 hours (13%) 
Assessment  58 hours (58%) 
Total  100 hours 
Private study description
Working on assignments, going over lecture notes, text books, exam revision.
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 D
Weighting  Study time  

Assignments  15%  20 hours 
Examination  85%  38 hours 

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100% 
Feedback on assessment
Marked homework (both assessed and formative) is returned and discussed in smaller classes. Exam feedback is given.
Courses
This module is Core for:
 Year 2 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)

UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of G100 Mathematics
 Year 2 of G103 Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Option list A for:
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)