|Semester 1 Credit Value:||10|
To introduce students to the basic ideas of group theory.
The course introduces the axioms of a group and explores many examples, particularly of symmetry groups, permutation groups, and matrix groups. Basic properties of groups will be demonstrated, always with reference to examples.
We shall prove Lagrange's Theorem, which tells us that for finite groups the number of elements in a subgroup divides the number of elements in the parent group. Groups of permutations will be studied systematically. We will introduce group homomorphisms and prove the Isomorphism Theorem which associates an isomorphism to each homomorphism. We also consider group actions, and Cayley's theorem and apply group actions to prove Cauchy’s and Sylow’s theorems, which are partial converses to Lagrange’s.
Symmetries. Definition of group. Groups of symmetries, groups of numbers, cyclic and dihedral groups, matrix groups. Subgroups, cosets, and Lagrange's Theorem. Symmetric and alternating groups. Isomorphisms, homomorphisms, quotient groups, and the Isomorphism Theorem. Group actions, Cauchy’s, Cayley's and Sylow’s theorems.
Group axioms; examples of groups of symmetries, numbers, matrices, permutations, subgroups, homomorphisms, isomorphisms, group actions, Lagrange's, Cauchy's, Cayley's and Sylow's theorems.
Perform calculations with permutations; be familiar with symmetric, alternating, cyclic, and dihedral groups; check group axioms, verify subgroups, homomorphisms; calculate orbits and stabilizers of group actions; compute the number of orbits; calculate cosets; know and use Lagrange's, Cauchy's, Cayley's and Sylow's theorems.
|Graduate Skills Framework Applicable:||Yes|
|Scheduled Learning And Teaching Activities||Lecture||22||1:00||22:00||Formal lectures|
|Guided Independent Study||Assessment preparation and completion||1||11:00||11:00||Revision for unseen Exam|
|Guided Independent Study||Assessment preparation and completion||1||1:30||1:30||Unseen Exam|
|Guided Independent Study||Assessment preparation and completion||5||5:00||25:00||Written assignments|
|Scheduled Learning And Teaching Activities||Lecture||2||1:00||2:00||Revision lectures|
|Scheduled Learning And Teaching Activities||Lecture||6||1:00||6:00||Problem classes|
|Scheduled Learning And Teaching Activities||Drop-in/surgery||6||1:00||6:00||Drop-ins in the lecture room|
|Guided Independent Study||Independent study||5||1:00||5:00||Assignment review|
|Guided Independent Study||Independent study||1||21:30||21:30||Studying, practising and gaining understanding of course material|
Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Drop-ins are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. Office hours provide an opportunity for more direct contact between individual students and the lecturer.
The format of resits will be determined by the Board of Examiners
|Prob solv exercises||1||M||10||N/A|
A substantial formal unseen examination is appropriate for the assessment of the material in this module. Written assignments (approximately 5 pieces of work of approximately equal weight) allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; this is thus formative as well as summative assessment.
Disclaimer: The information contained within the Module Catalogue relates to the 2016/17 academic year. In accordance with University Terms and Conditions, the University makes all reasonable efforts to deliver the modules as described. Modules may be amended on an annual basis to take account of changing staff expertise, developments in the discipline, the requirements of external bodies and partners, and student feedback. Module information for the 2017/18 entry will be published here in early-April 2017. Queries about information in the Module Catalogue should in the first instance be addressed to your School Office.