Lecturer: Richard Lissaman
Term(s): Term 1
Status for Mathematics students: Core
Commitment: 30 lectures, written assignments
Assessment: 15% from assignments and 85% from exam
Formal registration prerequisites: None
Assumed knowledge: Grade A in A-level Further Maths or equivalent
Useful background: Knowledge of modular arithmetic, manipulation of polynomials (including long division), language of functions, terms ‘commutative’ and ‘associative’
Leads to: The following modules have this module listed as assumed knowledge or useful background:
This module provides an introduction to two algebraic structures which are fundamental in mathematics: groups and rings. You'll meet both of these structures many times in your study of mathematics and they have many applications in other disciplines.
Both structures are concerned with combining objects in a collection to produce new objects in that collection, so called binary operations. Addition is an example of a binary operation on the integers. It takes in two integers e.g. 2 and 5 and gives us a new integer, 7 = 2 + 5.
We'll see that groups occur very naturally in mathematics, including mathematics you'll already be very familiar with, and that they are a means to classify the 'symmetries' of an object.
You are already very familiar with some examples of rings. The integers with their regular addition and multiplication is a ring. Rings have many applications including some in coding theory and cryptography.
To introduce groups and rings, driven by both examples and elementary theory.
Group Theory: motivating examples (numbers, cyclic group, dihedral group, symmetric group, transformations of the plane), elementary properties, subgroups, odd and even permutations.
Ring Theory: commutative and non-commutative rings, fields, examples (the integers, polynomials with integer coefficients, polynomials with real coefficients and quotient rings of such, unit groups, factorisations in the integers and polynomials.
List of covered algebraic definitions: group, subgroup, group homomorphism (including kernel, image, isomorphism), order, sign of permutation, ring, field, subring, ideal, ring homomorphism (including kernel, image, isomorphism), quotient ring
To explore a wide range of examples of groups and rings; to cover elementary properties of both structures; to start to look at ways to study and classify groups and rings as abstract objects. As a result of taking this module, students should be sufficiently prepared for the appearance of groups and rings in other first year courses and in follow-up second year courses.
Any book with ‘Abstract Algebra’ in the title is worth looking at (there are many). Here are a few specific recommendations (but note that comprehensive lecture notes will be provided):
Lara Alcock – How to Think About Abstract Algebra
Tony Barnard and Hugh Neill - Discovering Group Theory
I.N.Herstein - Topics in Algebra
Nathan Jacobson - Lectures in Abstract Algebra