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 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.

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.

Section 1 Group Theory
Motivating examples: numbers, symmetry groups
Definitions, elementary properties
Subgroups, including subgroups of Z
Arithmetic modulo n and the group Z_n
Lagrange's Theorem
Permutation groups, odd and even permutations (proof optional) Normal subgroups and quotient groups

Section 2 Ring Theory
Definitions: Commutative and non-commutative rings, integral domains, fields
Examples: Z, Q, R, C, Z_n, matrices, polynomials, Gaussian integers
Unit, unit groups, factorisation, examples where unique factorisation fails

Learning outcomes

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

• Understand the abstract definition of a group, and be familiar with the basic types of examples, including numbers, symmetry groups and groups of permutations and matrices.
• Understand what subgroups are, and be familiar with the proof of Lagrange’s Theorem.
• Understand the definition of various types of ring, and be familiar with a number of examples, including numbers, polynomials, and matrices.
• Understand unit groups of rings, and be able to calculate the unit groups of the integers modulo n.

Indicative reading list

Any library book with Abstract Algebra in the title would be useful.

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.