This module is not currently running in 2024 to 2025.
• Groups: revision, presentations of groups, Sylow's theorems and applications (e.g. classification of groups)
• Finitely generated abelian groups: finite abelian groups, Smith normal form, classification, applications (e.g. systems of linear Diophantine equations)
• Knots: introduction, Reidemeister moves, knot invariants, the Abelian knot group
• Fields: revision, soluble groups, Galois Theorem, applications (e.g. impossibility of solving the quintic)
In addition, for level 7 students:
• Advanced topic such as proof of the Galois Theorem, the Jones polynomial, the Alexander polynomial, braid groups or Polya enumeration.
Total contact hours: 42
Private study hours: 108
Total study hours: 150
80% examination, 20% coursework
Group theory:
M. Aschbacher: Finite Group Theory (Cambridge Studies in Advanced Mathematics), Cambridge University Press, 2000
B. Baumslag and B. Chandler: Schaum's Outline of Group Theory, McGraw Hill Professional, 1968
A. Kerber, Applied Finite Group Actions, Springer, 1999
Knot theory:
C. Livingston, Knot theory, Mathematical Association of America, 1993
V. Manturov, Knot Theory, Chapman & Hall, 2004
Field theory:
John M. Howie, Fields and Galois Theory, Springer, 2006
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of Groups, Knots and Fields ;
2 demonstrate the capability to solve complex problems using a very good level of skill in calculation and manipulation of the material in the following areas: groups, Sylow's Theorems, finitely generated abelian groups, Smith normal form, knots and their invariants, Galois extensions;
3 apply a range of concepts and principles in Groups, Knots and Fields theory in loosely defined contexts, showing good judgment in the selection and application of tools and techniques.
The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make effective use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material effectively;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.
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