Course Outline

Groups,Rings & Modules

Basic concepts of group theory recalled from Part IA Groups. Normal subgroups, quotient groups and isomorphism theorems. Permutation groups. Groups acting on sets, permutation representations.
Conjugacy classes, centralizers and normalizers. The centre of a group. Elementary properties of finite p-groups. Examples of finite linear groups and groups arising from geometry. Simplicity of A_n

Definition and examples of rings (commutative, with 1). Ideals, homomorphisms, quotient rings, iso-

morphism theorems. Prime and maximal ideals. Fields. The characteristic of a field. Field of fractions

of an integral domain.
Factorization in rings; units, primes and irreducibles. Unique factorization in principal ideal domains,

and in polynomial rings. Gauss’ Lemma and Eisenstein’s irreducibility criterion.
Rings Z[alpha] of algebraic integers as subsets of C and quotients of Z[x]. Examples of Euclidean domains and uniqueness and non-uniqueness of factorization. Factorization in the ring of Gaussian integers;representation of integers as sums of two squares
Ideals in polynomial rings. Hilbert basis theorem.

Definitions, examples of vector spaces, abelian groups and vector spaces with an endomorphism. Submodules, homomorphisms, quotient modules and direct sums. Equivalence of matrices, canonical form.

Structure of finitely generated modules over Euclidean domains, applications to abelian groups and

Jordan normal form

P.J. Cameron Introduction to Algebra. OUP

J.B. Fraleigh A First Course in Abstract Algebra. Addison Wesley, 2003

B. Hartley and T.O. Hawkes Rings, Modules and Linear Algebra: a further course in algebra. Chapman and Hall, 1970

I. Herstein Topics in Algebra. John Wiley and Sons, 1975

P.M. Neumann, G.A. Stoy and E.C. Thomson.Groups and Geometry. OUP 1994