Introductory description

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

Module web page

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.

• Groups, subgroups and normal subgroups, rings and fields.
• Isomorphisms, homomorphisms, direct products and quotients for both groups and rings.
• Examples of groups including cyclic, dihedral, symmetric and alternating groups, additive and multiplicative groups of a ring, and general linear groups over a field.
• Generators and relations for groups.
• Order of an element, cosets, Lagrange's theorem, Euler's theorem.
• First, second and third isomorphism theorems and the correspondence theorem for groups.
• Group actions, the Orbit-Stabiliser theorem, conjugacy classes and centralisers.
• Conjugacy classes in symmetric and alternating groups, simplicity of A5.
• Chinese Remainder Theorem.
• First isomorphism theorem and correspondence theorem for rings.
• Ideals, maximal ideals, domains and field of fractions.
• Unique factorisation domains, principal ideal domains, prime and irreducible elements.
• Factorisation in polynomial rings, remainder theorem for polynomials over a field, Gauss's Lemma and Eisenstein's criterion.

Learning outcomes

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

• Recall definitions, theorems and proofs of syllabus material. Calculate with examples of groups and rings and solve problems using algebraic techniques. Recognise, classify and construct examples of groups and rings with specified properties by applying the concepts listed in the syllabus. Apply the orbit-stabiliser theorem to conjugacy classes and centralisers, as well as to examples from number theory, geometry and combinatorics.

Indicative reading list

Niels Lauritzen, Concrete Abstract Algebra, Cambridge University Press.

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.