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Next: MC342 History of mathematics
Up: Year 3
Previous: MC316 Parallel and Distributed Computing
MC341 Group Theory
Credits: 20 |
Convenor: Dr. R. Marsh |
Semester: 2 |
Prerequisites: |
essential: MC242 |
|
Assessment: |
Coursework: 10% |
Three hour exam: 90% |
Lectures: |
36 |
Problem Classes: |
10 |
Tutorials: |
none |
Private Study: |
104 |
Labs: |
none |
Seminars: |
none |
Project: |
none |
Other: |
none |
Surgeries: |
none |
Total: |
150 |
Explanation of Pre-requisites
This module builds on the module MC242, Introduction to Groups, by developing
the theory of groups; therefore it is essential that students are familiar
with the content of MC242, although a brief revision of the material from
MC242 will take place at the start of the course.
Course Description
This module builds upon MC242 (Introduction to Groups) and gives an overview
of the main ideas of group theory. The emphasis is on structure theorems,
classification results and decomposition concepts that have evolved as a
result of attempts to describe all possible groups.
Although such a description is not actually feasible, it is possible to
obtain surprisingly detailed information about the structure of large
classes of groups.
One of the major goals of the module is to develop enough theory to be able
to discuss the classification of the finite simple groups, at least in broad
general terms. This classification ranks as one of the major achievements in
pure mathematics. Its proof runs to somewhere between 10,000 and 15,000
journal pages, spread across some 500 separate articles by more than 100
mathematicians, almost all written between 1950 and the early 1980's.
A revision is currently underway, but even this is expected to run to more
than 5,000 pages.
The module is designed to present a broad outline of the classification, to
explain its significance, and to give a hint of the complexity of its proof.
Aims
This course aims to present the fundamental ideas of group theory by studying
the structure theorems and decomposition concepts that arise in attempts to
understand groups in terms of less complicated groups. These attempts are
most successful in studying finite groups because there is a sense in which
any finite group can be regarded as a group built from finite simple groups.
The course aims to develop the ideas necessary to make this notion precise
and to develop the theory needed to present a rough idea of the statement of
the classification of finite simple groups.
Objectives
By the end of this module students should have developed an understanding of
- the concepts of homomorphisms, normal subgroups and quotient groups
and their relevance to the structure of a group;
- the idea of a group action and how group actions are used for
enumeration and to prove fundamental results such as the Sylow theorems;
- the idea of a group presentation and how to calculate using generators
and relations;
- the properties of permutations and of symmetric and alternating groups;
- the idea of simple groups as the basic building blocks of group theory.
- composition series and chief series for groups, the
Jordan-Hölder Theorem and solvable and nilpotent groups.
Transferable Skills
This module should help the student develop a good sense of the axiomatic
approach to mathematics. In addition, it provides students with practice in
presenting reasoned arguments with precision and cogency.
Syllabus
Definition and examples of groups; the symmetric group and cycle
decomposition of elements; free groups; subgroups; centralisers; centre of
a group; products of subsets; cosets; Lagrange's Theorem; conjugates and
conjugacy classes of elements and of subsets; normal subgroups;
normalisers; quotient groups; homomorphisms and isomorphisms;
kernels and images; the isomorphism theorems; normal closures;
presentations of groups; generators and relations; direct products;
automorphism groups and the conjugation map;
G-sets and actions; the kernel of an action; faithful and transitive
actions; multiplication and conjugation actions; stabilisers and orbits; the
Orbit-Stabiliser Theorem; the fixed point theorem for the number of orbits
of an action; use of this theorem for enumeration.
the alternating groups An; Cayley's theorem that every group is isomorphic
to a subgroup of a symmetric group;
Sylow subgroups of finite groups; the Sylow theorems;
characteristic subgroups; simple groups; normal and subnormal series;
composition series and chief series; the Jordan-Hölder Theorem;
solvable and nilpotent groups; commutator subgroups;
upper and lower central series; derived series; that a finite p-group is
nilpotent; normalisers in nilpotent groups; characterisation of nilpotent
groups; simplicity of the alternating groups An for
;outline of the classification of finite simple groups.
Reading list
Recommended:
J. F. Humphreys,
A Course in Group Theory,
Oxford University Press.
Background:
J. A. Gallian,
Contemporary Abstract Algebra,
DC Heath.
D. Gorenstein, R. Lyons and R. Solomon,
The classification of the finite simple groups,
Mathematical Surveys and
Monographs, 40.1. American Mathematical Society.
D. L. Johnson,
Presentations of Groups,
Cambridge University Press.
C. R. Jordan and D. A. Jordan,
Groups,
Edward Arnold.
I. D. MacDonald,
Theory of Groups,
Oxford University Press.
J. S. Rose,
A Course in Group Theory,
Cambridge University Pres.
J. J. Rotman,
An Introduction to the Theory of Groups,
Springer-Verlag.
Details of Assessment
There will be around 8 pieces of work set for assessment which will
together count for 10% of the final mark; the final examination contributes
the remaining 90%.
The exam paper contains 8 questions. Any number of questions may be
attempted, but only the best 5 answers will be taken into account. Full
marks may be obtained for answers to 5 questions. All questions carry
equal weight.
Next: MC342 History of mathematics
Up: Year 3
Previous: MC316 Parallel and Distributed Computing
Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 10/4/2000
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This document has been approved by the Head of Department.
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