Introductory description

This is an introductory abstract algebra module. As the title suggests, the two main objects of study are groups and rings.

Module aims

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: motivating examples (numbers, dihedral group, quaternionic group, matrix groups), elementary properties, subgroup, coset, Lagrange’s theorem, quotient group, isomorphism theorem, free group, group given by generators and relations, group action, G-set G/H, orbit, stabiliser, the orbit-stabiliser theorem, conjugacy class, classes in S_n, classification of groups up to order 8.
• Ring Theory: commutative and non-commutative ring, domain, examples (Z[x], Z/nZ, F[x], F[x]/(f)), ideal, quotient ring, 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, Eisenstein criterion, Gauss lemma, cyclotomic polynomial, finite subgroups 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: group or ring homomorphism (including kernel, image, isomorphism), 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.

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
• get the working knowledge of the understand the definition of various types of ring, and be familiar with a number of examples, including numbers, polynomials and Z/nZ

Indicative reading list

Samir Siksek, Introduction to Abstract Algebra lecture notes,
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.