MA301-7-SP-CO:
Group Theory
2022/23
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Postgraduate: Level 7
Current
Monday 16 January 2023
Friday 24 March 2023
15
29 March 2022
Requisites for this module
(none)
(none)
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Groups are essential to abstract algebra. Indeed many algebraic structures (including rings, fields, modules of rings and vector spaces) are groups with extra operations. Groups are not only key to understand other areas of mathematics such as geometry and representation theory, they have also influenced our understanding of other sciences including physics, chemistry, material science or cryptography.
The idea of a group can be perhaps best explained in a simple example: the symmetries of a triangle. The equilateral triangle have 6 symmetries: three rotations (120, 240 and 360 degrees) and three reflections (each of them along the line passing by a vertex and the middle of the opposite edge). Because these six symmetries form a group, the composition of any two of them give another symmetry in the group. Furthermore, each symmetry can be reverted by another symmetry. In group theory the real numbers and the symmetries of the equilateral triangle can be treated uniformly through axioms and theorems that derive from these axioms. Ultimately, some of the theorems of this module will allow us to classify groups under certain assumptions. More generally, the aims of the course are:
1. To introduce students to basic definitions of group theory and representation theory.
2. To develop students’ critical understanding of some main results in group theory and representation theory.
3. To develop understanding of how to apply such results in particular problems.
On completion of the course, students should:
1. Have a systemic understanding of key definitions in the theory of groups and critical awareness of how they interact and support each other.
2. Select and apply relevant theorems to examples.
3. Construct arguments to prove properties of groups.
4. Solve problems involving homomorphisms between pairs of groups.
5. Formulate counterexamples to statements.
6. Understand the concept of a group presentation.
7. Recognise and work with cyclic, dihedral, symmetric and alternating groups.
8. Deploy methods learned to distinguish pairs of groups defined by presentations or to prove they are isomorphic.
9. Apply geometric techniques to obtain and illustrate algebraic properties of particular groups.
10. Understand some operations to construct groups from other groups, including direct product, semiproduct product, abelianisation and derived subgroups.
11. Understand the notion of representation of a group and being able to describe irreducible representations in some simple examples.
Syllabus
Homomorphisms, isomorphisms, kernel and image, automorphisms.
Cosets, normal subgroups and quotient groups. First isomorphism theorem. Conjugacy classes and centraliser of a group.
Symmetry group and cycle structure. Alternating group. Cayley's theorem. Dihedral groups.
Direct and semi-direct products. Classification of finite abelian groups.
Free groups, group presentations (definition and examples).
Group actions and orbit-stabiliser theorem.
Abelianisation and derived subgroup.
Cayley Diagrams.
Introduction to Representation Theory. Schur's lemma
Teaching in the department will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.
This module does not appear to have a published bibliography for this year.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Assignment |
|
|
| Exam |
Main exam: In-Person, Closed Book, 120 minutes during Summer (Main Period)
|
| Exam |
Reassessment Main exam: In-Person, Closed Book, 120 minutes during September (Reassessment Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Jesus Martinez-Garcia, email: jesus.martinez-garcia@essex.ac.uk.
Dr Jesus Martinez-Garcia
jesus.martinez-garcia@essex.ac.uk
Yes
No
No
Prof Stephen Langdon
Brunel University London
Professor
Dr Rachel Quinlan
National University of Ireland, Galway
Senior Lecturer in Mathematics
Available via Moodle
Of 38 hours, 38 (100%) hours available to students:
0 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s), module, or event type.
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