- Department: Mathematics
- Credit value: 20 credits
- Credit level: H
- Academic year of delivery: 2023-24
- See module specification for other years: 2024-25
Groups measure the symmetry of concrete and abstract objects. This course continues their story from second year with an emphasis on the concept of a group action. The second half of the course links group theory with ring theory and shows how groups can be used to measure the symmetries of polynomial equations.
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Occurrence | Teaching period |
---|---|
A | Semester 1 2023-24 |
Groups measure the symmetry of concrete and abstract objects. This course continues their story from second year with an emphasis on the concept of a group action. The second half of the course links group theory with ring theory and shows how groups can be used to measure the symmetries of polynomial equations.
By the end of the module, students will be able to:
Apply group actions to numerous topics in group theory.
Construct Sylow subgroups and apply the Sylow theorems to problems involving simple groups.
Construct the various kinds of field extensions.
Use the Galois correspondence to analyse intermediate fields.
Group actions, orbits, stabilisers and the orbit-stabiliser theorem.
Cauchy’s theorem, counting orbits and Burnside’s theorem.
Homomorphism theorems
Sylow theorems.
Finite simple groups
Conjugacy.
Fields as quotients of polynomial rings.
Field extensions, splitting fields.
Constructible numbers.
The Galois group of an extension.
The Galois correspondence and some applications.
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
None
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
M A Armstrong, Groups and Symmetry, Springer UTM.
Ian Stewart, Galois Theory, Routledge.