MT381 Algebra III (given in 2007-8)

Prerequisite: MT282 and MT283

Teaching: 33hr lectures

Assessment: 2hr written examination

Aims

The aim of this course is to give the students a working knowledge of finitely-generated abelian groups and the analogous theory of modules.

Learning outcomes

On completion of the course, students should be able to:

· demonstrate the existence of normal subgroups of certain finite groups, and to use this to determine the isomorphism classes of groups of certain orders;

· classify all finitely-generated abelian groups;

· understand the concepts of module, submodule and lattice;

· prove that the minimum polynomial of a matrix is invariant under similarity, but not conversely;

· prove that A is similar to the companion matrix of its characteristic polynomial if and only if the corresponding module is cyclic;

· obtain the rational canonical form, primary canonical form or Jordan form as appropriate.

Content

Group Theory: Homomorphisms, normal subgroups, factor groups, the first isomorphism theorem. The centre, centralizers, groups of order .

Modules: The –module determined by a square matrix over F, the order of a module element, cyclic submodules, companion matrices, minimum polynomials. Abelian groups as Z–modules.

Module decomposition: Direct sum, primary decomposition. Free R-modules and their rank, invertible matrices over R. Equivalent matrices over R, reduction to invariant factor form (Smith normal form) over and Z. The rational, primary and Jordan forms of a square matrix over F. The classification of finitely–generated abelian groups.

Indicative Text

Rings, Modules and Linear Algebra - B Hartley & T Hawkes (Chapman & Hall). Library Ref. 512.61 HAR

Topics in Algebra – I N Herstein (Wiley 1975) Library Ref. 512.61 HER