Introductory description

This module introduces important algebraic structures including groups, rings and fields. Students will learn how to verify that a set is a group, ring or field, and how to carry out elementary operations in these structures. They will also assimilate permutations, symmetric groups, and alternating groups, and know how to determine the unit group of a ring.

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, cyclic group, dihedral group, symmetric group, transformations of the plane), elementary properties, subgroups, Lagrange's Theorem for abelian groups, odd and even permutations.
• Ring Theory: commutative and non-commutative rings, fields, examples (Z[x], Z/nZ, F[x], F[x]/(f)), unit groups, factorisations in Z 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

Learning outcomes

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

• understand the abstract definition of a group and a group homomorphism
• be familiar with the dihedral and cyclic groups as well as the group of euclidean transformations of the plane
• perform manipulations with the elements of the symmetric group, representing them as a product of compositions
• understand the orders of elements as well as the proof of Lagrange’s Theorem for abelian groups
• 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
• perform manipulations with polynomials over Z, R and C
• learn the unit groups of rings, in particular, of 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

Subject specific skills

This module introduces important algebraic structures including groups, rings and fields. Students will learn how to verify that a set is a group, ring or field, and how to carry out elementary operations in these structures. They will understand the relation between a group, a subgroup and the cosets of a subgroup which leads to Lagrange's theorem. They will also assimilate permutations, symmetric groups, and alternating groups, and know how to determine the unit group of a ring.

Transferable skills

The module reinforces logical thinking and deductive reasoning which are valuable transferable skills. The algebraic structures introduced are the heart of modern cryptography and information security.