UNIVERSITY OF KENT
Confirmation that this version of the module specification has been approved by the School
Learning and Teaching Committee:
…13th May 2015…….…………………….(date)
SECTION 1: MODULE SPECIFICATION
1.
Title of the module: Symmetries, Groups and Invariants (MA605 and MA965)
2.
School or partner institution which will be responsible for management of the module: School of
Mathematics, Statistics and Actuarial Science
3.
Start date of the module: MA605 September 2015; MA965 October 2010 (revised version start date
September 2015)
4.
The number of students expected to take the module: 25
5.
Modules to be withdrawn on the introduction of this proposed module and consultation with other
relevant Schools and Faculties regarding the withdrawal: MA965 is an existing M-level module. A
new H-level version of this module, MA605, is being created. No consultation with other Schools is
necessary.
6.
The level of the module (e.g. Certificate [C], Intermediate [I], Honours [H] or Postgraduate [M]):
MA605 – H level; MA965 – M level.
7.
The number of credits and the ECTS value which the module represents: 15 (ECTS 7.5)
8.
Which term(s) the module is to be taught in (or other teaching pattern): Autumn or Spring term
9.
Prerequisite and co-requisite modules:
For the H–level module, the prerequisite modules are MA553 Linear Algebra and MA565 Groups
and Rings. There are no co-requisite modules.
For the M–level module, there are no prerequisite modules although students are expected to have
studied introductory courses on linear algebra and groups. There are no co-requisite modules.
10. The programmes of study to which the module contributes:
For the H–level module: BSc (Hons) Mathematics (including programme with a year in industry), BSc
(Hons) Mathematics with a Foundation Year, MMath Mathematics, Graduate Diploma in
Mathematics
For the M–level module: International MSc Mathematics and its Applications, MSc in Mathematics
and its Applications, MMath Mathematics.
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UNIVERSITY OF KENT
For the MMath Mathematics and MMathStat Mathematics and Statistics programmes, either the Hlevel or the M-level module may be taken, but not both.
11. The intended subject specific learning outcomes
On successful completion of this module, H–level students will have increased their knowledge,
understanding, intuition and computational expertise in:
(a) detecting symmetries and common patterns;
(b) using group theory to calculate with symmetries;
(c) the distinction and classification of objects up to equivalences and symmetries;
(d) the use of “normal forms” and “invariants” to distinguish symmetry classes.
They will also have:
(e) an enhanced ability to correctly formulate classification problems and solve them efficiently;
(f) an appreciation of algorithms and computational methods in algebra and group theory;
(g) consolidated a variety of tools from abstract algebra to model and classify concrete objects
and configurations.
On successful completion of this module, M–level students will also
(h) have learned finer methods to count with symmetries as applied in the sciences (e.g. counting
dioxines, Polya theory).
12. The intended generic learning outcomes
On successful completion of this module, H–level students will have:
(a) an increased ability in rigorous thinking;
(b) increased their understanding of systematic observation, generalization and techniques of
proof;enhanced skills in understanding and communicating mathematical results and
conclusions;
(c) a holistic view of mathematics as a problem solving and intellectually stimulating discipline;
(d) matured in their problem formulating and solving skills.
On successful completion of this module, M–level students will also have
(e) an enhanced ability to read independently and comprehend mathematical ideas from a text
and to manage their time.
13. A synopsis of the curriculum
In this module we will study certain configurations with symmetries as they arise in real world
applications. Examples include knots described by “admissible diagrams” or chemical structures
described by “colouring patterns”. Different diagrams and patterns can describe essentially the same
structure, so the problem of classification up to equivalence arises. This will be solved by attaching
“invariants” which are then put in “normal form” to distinguish them. The syllabus will be as follows:
(a) review of basic methods from linear algebra, group theory and discrete mathematics;
(b) permutation groups, transitivity, primitivity, Burnside formula;
(c) finitely generated Abelian groups;
(d) applications to knot theory, Reidemeister moves, the Abelian knot group;
(e) examples, observations, generalizations and proofs.
In addition, for M–level students,
(f) General Polya-enumeration (as an extension of the Burnside formula).
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14. Indicative Reading List
Group theory:
(a)
(c)
M. Aschbacher: Finite Group Theory (Cambridge Studies in Advanced
Mathematics), Cambridge University Press, 2000, (H-level/M-level)
B. Baumslag and B. Chandler: Schaum's Outline of Group Theory, McGraw Hill
Professional, 1968, (H-level)
A. Kerber, Applied Finite Group Actions, Springer, 1999. (M-level)
(d)
(e)
C. Livingston, Knot theory, Mathematical Association of America, 1993, (H-level)
V. Manturov, Knot Theory, Chapman & Hall, 2004, (M-level).
(b)
Knot theory:
15. Learning and Teaching Methods, including the nature and number of contact hours and the total
study hours which will be expected of students, and how these relate to achievement of the intended
module learning outcomes
Total contact time: 42-48, including approx. 36 lectures and 6-12 example classes (1-hour).
Number of independent learning hours: 102-108 hours
Total study time: 150 hours.
Lectures address learning outcomes 11(a)-(g) and 12(a)-(e); classes and/or workshops address
11(d), 12(a)-(e).
In addition, for M–level students, critical analysis of directed reading is used to address learning
outcomes 11(h) and 12(f).
16. Assessment methods and how these relate to testing achievement of the intended module learning
outcomes
Assessment: The module is assessed on the basis of examination (80%) and coursework (20%).
Coursework will assess learning outcomes 11(a)-(g) and 12(a)-(e). Different open-book assignments
will be provided for M-level students, which will additionally assess learning outcomes 11(h) and
12(f). A 3-hour written examination will assess learning outcomes 11(a)-(e) and 12(a)-(c),(e). In
each case, questions may involve elements of bookwork and solving mathematical problems. The
examination set for M-level students will assess their level of understanding to an enhanced level..
17. Implications for learning resources, including staff, library, IT and space:
For the lectures and the workshops, a room with plenty of white-board space is required.
18. The School recognises and has embedded the expectations of current disability equality legislation,
and supports students with a declared disability or special educational need in its teaching. Within
this module we will make reasonable adjustments wherever necessary, including additional or
substitute materials, teaching modes or assessment methods for students who have declared and
discussed their learning support needs. Arrangements for students with declared disabilities will be
made on an individual basis, in consultation with the University’s disability/dyslexia support service,
and specialist support will be provided where needed.
19.
Campus where module will be delivered: Canterbury
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Module Specification Template (v.October 2014)