MA301-6-SP-CO:
Group Theory
2024/25
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Undergraduate: Level 6
Current
Monday 13 January 2025
Friday 21 March 2025
15
03 May 2024
Requisites for this module
MA114 and MA201 and MA204
(none)
(none)
(none)
(none)
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 aims of this module are:
- To introduce students to basic definitions of group theory
- To develop students’ critical understanding of some main results in group theory
- To develop understanding of how to apply such results in particular problems.
By the end of the module, students will be expected to be able to:
1. Formulate and evaluate statements involving groups and their properties.
2. Recognise and work with various common examples of groups, such as cyclic, dihedral, symmetric and alternating groups and matrix groups.
3. Select and apply relevant theorems to examples.
4. Prove basic results involving abstract groups and their properties, and construct counterexamples to statements.
5. Understand standard constructions in group theory, such as group actions, group presentations, conjugacy, homomorphisms, isomorphisms, quotients and products.
Group theory is the branch of abstract algebra that formalises the rules obeyed by collections of symmetries, and other sets with rules for combining elements, such as various types of numbers with addition or multiplication operations. From a few basic axioms, an extraordinarily rich structure theory emerges, with applications across all of mathematics and the sciences. Common basic questions then include: Can we classify certain types of groups? And what mathematical results apply to all groups?
Indicative syllabus
- Homomorphisms, isomorphisms, automorphisms; kernels and images
- Cosets, normal subgroups and quotient groups. First isomorphism theorem.
- Conjugacy classes and centralisers.
- Symmetric and alternating groups; cycle structure; Cayley's theorem.
- Direct and semi-direct products.
- Classification of finite abelian groups.
- Free groups and group presentations (definition and examples).
- Group actions and orbit-stabiliser theorem
- Abelianisation and derived subgroup.
- Cayley graphs.
Teaching in the School 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.
The above list is indicative of the essential reading for the course.
The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.
Further reading can be obtained from this module's
reading list.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Test |
|
|
| 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
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 33 hours, 33 (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).
* Please note: due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.
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