MATH30002/MATH40002 Group TheoryThis is the former 312/MA3002 unit SEMESTER: First CONTACT: Professor Peter Rowley (M/O14) CREDIT RATING: 10
 Aims: To introduce students to some more sophisticated concepts and results of group theory as an essential part of general mathematical culture and as a basis for further study of more advanced mathematics. Intended Learning Outcomes: On successful completion of the course students will have acquired: A sound understanding of the classification of finitely generated abelian groups,  Knowledge of some fundamental results and techniques from the theory of finite groups. Knowledge of group actions on sets, simple groups, Sylow's theorems and various applications of Sylow's theorems. Pre-requisites: 112, 152, 212, 252 (ex-UMIST), MT2262 (ex-VUM) Dependent Courses: MATH40005, MATH 40012 Course Description: This course is a continuation of 212, the Second Year Group Theory course. The ideal aim of Group Theory is the classification of all groups (up to isomorphism). It will be shown that this goal can be achieved for finitely generated abelian groups. In general, however, there is no hope of a similar result as the situation is far too complex, even for finite groups. Still, since groups are of great importance for the whole of mathematics, there is a highly developed theory of outstanding beauty. It takes just three simple axioms to define a group, and it is fascinating how much can be deduced from so little. The course is devoted to some of the basic concepts and results of Group Theory. Teaching Mode: 2 Lectures per week 1 Tutorial per week Private Study: 5 hours per week Recommended Text: John B Fraleigh, A First Course in Abstract Algebra, (5th edition), 1967, Addison-Wesley. Assessment Methods: Coursework: 10% (MATH30002), 7% (MATH40002) Examination: 90% (MATH30002), 93% (MATH40002) Examination is of 2 hours duration (MATH30002), 3 hours duration (MATH40002) at the end of the First Semester.
 No. of lectures Syllabus 2 Revision of basic notions (subgroups and factor groups, homomorphisms and isomorphisms), generating sets, commutator subgroups. 4 Abelian groups, the Fundamental Theorem on finitely generated abelian groups. 3 The Isomorphism Theorems. 3 Simple groups, the simplicity of the alternating groups. 2 Composition series, the Jordan-Hölder Theorem. 5 Group actions on sets, orbits, stabilizers, the number of elements in an orbit, Burnside's formula for the number of orbits, conjugation actions, centralizers and normalizers. 3 Sylow's Theorems, groups of order pq, pqr. For MATH40002 the lectures will be enhanced by additional reading on related topics.

Last revised August, 2006