2024/25 Taught Postgraduate Module Catalogue

MATH5071M Groups, Symmetry and Galois Theory

20 creditsClass Size: 60

Module manager: Professor Kevin Houston
Email: K.Houston@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2024/25

Pre-requisite qualifications

(MATH2020 or MATH2022, or equivalent) and (MATH2025 or MATH2026 or MATH2027, or equivalent).

This module is mutually exclusive with

MATH3071	Groups and Symmetry
MATH5246M	Fields and Galois Theory

Module replaces

MATH5246M

This module is not approved as an Elective

Module summary

Group theory is the mathematical theory of symmetry. Groups arise naturally in both pure and applied mathematics, for example in the study of permutations of sets, rotations and reflections of geometric objects, symmetries of physical systems and the description of molecules, crystals and materials. Groups have beautiful applications to counting problems, in which objects are counted up to symmetry, answering questions like: “How many ways are there to colour the faces of a cube with m different colours, up to rotation of the cube?”.As in the case of quadratic equations, degree 3 and 4 polynomial equations also have formulas for their solution. However, there is no analogous formula for the roots of a general polynomial equation of degree 5 or more. The proof of this fact requires Galois theory, which is the study of the way in which one field can be contained in another. A key role is played by the Galois group.Many mathematicians regard Galois theory as one of the most beautiful theories in mathematics, since it brings together ideas from a number different areas of mathematics: groups, rings, fields and polynomials, which all interact while achieving strong fundamental results.

Objectives

To review and develop the basic notions and theorems of group theory.
To introduce the notion of a group acting on a set and its properties.
To study the powerful Sylow theorems which give information on the structure of an arbitrary finite group in terms of the prime divisors of its order; to use group actions in this study.
To give an introduction to Polya counting theory and how it is used to count objects up to symmetry.
To introduce the Galois theory of a field extension and its application to the existence of formulae giving the roots of polynomial equations.
To show how the Galois group of a field extension plays a role in Galois theory.
To develop the skills of rigorous logical argument and problem-solving in the context of group theory, symmetry and Galois theory.

Learning outcomes
On completion of this module, students should be able to:
a) Prove and use basic results on groups, homomorphisms and quotients;
b) Prove and use basic results on group actions;
c) Represent a group by permutations;
d) Use Sylow’s theorems to show that a group is not simple;
e) Apply Polya counting theory to simple counting problems;
f) Determine whether a group is soluble;
g) Compute the Galois group of a small field extension.

Syllabus

Revision of basic properties of groups.
Subgroups, homomorphisms, Lagrange's Theorem
Symmetric group, sign of a permutation, cycle decomposition
Isomorphism theorems
Group actions, Cayley's Theorem, Orbit-Stabiliser Theorem, application to symmetric group.
Sylow theorems and applications, simple groups
Burnside's Lemma, Polya counting theory
Soluble groups
Field extensions and splitting fields
Galois group of a polynomial
Solubility by radicals and relationship to soluble groups
An insoluble degree 5 polynomial
If time permits, extra topics to be chosen from: group presentations, symmetry groups of planar and three-dimensional figures, ruler and compass constructions.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	44	1.00	44.00
Private study hours			156.00
Total Contact hours			44.00
Total hours (100hr per 10 credits)			200.00

Private study

Study and revision of course material.

Completion of assignments and assessments.

Opportunities for Formative Feedback

Regular problem sheets.

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	3 hr 00 mins	100.00
Total percentage (Assessment Exams)		100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 08/10/2024

Disclaimer

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