# 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: **

- 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:**

- MA257 Introduction to Number Theory
- MA3E1 Groups and Representations
- MA3G6 Commutative Algebra
- MA3J9 Historical Challenges in Mathematics
- MA377 Rings and Modules
- MA3F1 Introduction to Topology
- MA3K4 Introduction to Group Theory
- MA3J3 Bifurcations, Catastrophes and Symmetry
- MA3D5 Galois Theory
- MA3H6 Algebraic Topology
- MA3J2 Combinatorics II
- MA3A6 Algebraic Number Theory
- MA4L6 Analytic Number Theory
- MA4H4 Geometric Group Theory
- MA426 Elliptic Curves
- MA473 Reflection Groups
- MA453 Lie Algebras
- MA4J8 Commutative Algebra II
- MA4M6 Category Theory

**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)