# MA267 Groups and Rings

Lecturer: Nicholas Jackson

Term(s): Term 1

THIS MODULE IS NOT AVAILABLE TO MATHS (G100/G103) STUDENTS

Commitment: 30 one-hour lectures plus assignments

Assessment: 85% by 2-hour examination, 15% coursework

Formal registration prerequisites: None

Assumed knowledge:

MA138 Sets and Numbers

• Number theory: congruence modulo-n, prime factorisation, Euclidean algorithm, greatest common divisors (gcd) and least common multiples (lcm).
• Sets and functions: basic set theory, injective and surjective functions, equivalence relations.
• Polynomials: multiplication and division, Euclidean algorithm, Remainder Theorem.

Synergies:

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content: This is an introductory abstract algebra module. As the title suggests, the two main objects of study are groups and rings. A group is a set with one binary operation; examples include the additive group of integers, groups of permutations, and groups of nonsingular matrices. Rings are sets with two binary operations, analogous to addition and multiplication. The most familiar example is the ring of integers with the usual addition and multiplication operations, and others include rings of polynomials, and rings of square matrices.

This module will assume no prior knowledge of the subject, but students who have previously taken MA151 Algebra 1 will have met some of the basic concepts already.

Topics covered:

• Group theory: Basic definitions and properties of groups, subgroups and homomorphisms. Cosets and Lagrange's Theorem. Normal subgroups and quotient groups. Cyclic groups, permutation groups, dihedral groups. Isomorphism theorems. Group actions, orbits and stabilisers, conjugacy classes, simple groups. Classification of finitely-generated abelian groups.
• Ring theory: Basic definitions and properties of rings, subrings and homomorphisms. Ideals and quotient rings. Integral domains, Euclidean domains, Principal Ideal Domains (PIDs), Unique Factorisation Domains (UFDs). Prime and irreducible elements. Fields. Polynomial rings.

Objectives: By the end of the module, the student should have a good working knowledge of the basic concepts of group theory and ring theory, and be familiar with a number of standard theorems and techniques.

Books:

• M A Armstrong, Groups and Symmetry, Undergraduate Texts in Mathematics, Springer
• John B Fraleigh, A First Course in Abstract Algebra, Pearson
• Joseph Gallian, Contemporary Abstract Algebra, Chapman Hall
• Nicholas Jackson, A Course in Abstract Algebra, Oxford University Press (forthcoming, draft sections available on request)