This is the former
Professor Peter Rowley (M/O14)
||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.
||On successful completion of the course students will have
- A sound understanding of the classification of finitely generated abelian groups,
- Knowledge of some fundamental results and techniques from the theory of finite
- Knowledge of group actions on sets,
simple groups, Sylow's theorems and various applications of Sylow's
||112, 152, 212,
252 (ex-UMIST), MT2262 (ex-VUM)
||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
||2 Lectures per week
||1 Tutorial per week
||5 hours per week
||John B Fraleigh, A First Course in Abstract Algebra, (5th
edition), 1967, Addison-Wesley.
(MATH30002), 7% (MATH40002)
(MATH30002), 93% (MATH40002)
||Examination is of 2 hours duration
(MATH30002), 3 hours duration (MATH40002) at the end of the First
|No. of lectures
||Revision of basic notions (subgroups and factor groups,
homomorphisms and isomorphisms), generating sets, commutator subgroups.
||Abelian groups, the Fundamental Theorem on finitely generated
||The Isomorphism Theorems.
||Simple groups, the simplicity of the alternating groups.
||Composition series, the
||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.
||Sylow's Theorems, groups of order
MATH40002 the lectures will be enhanced by additional reading on related