MMath Mathematics with Financial Mathematics

This course unit provides an introduction to groups, one of the most important algebraic structures. It gives the main definitions, some basic results and a wide range of examples. This builds on the study of topics such as properties of the integers, modular arithmetic, and permutations included in MATH10101/MATH10111. Groups are a fundamental concept in mathematics, particularly in the study of symmetry and of number theory.

The course unit unit aims to introduce basic ideas group theory with a good range of examples so that the student has some familiarity with the fundamental concepts of abstract algebra and a good grounding for further study.

On completion of this unit successful students will be able to: 

- state the group axioms and identify frequently met examples of groups,
- define basic concepts in group theory, such as subgroups, conjugacy classes, cyclic groups, cosets, and factor groups,
- employ the subgroup criterion to determine whether certain subsets of a group are subgroups,
- describe fundamental properties of cosets and factor groups,
- identify the generators and subgroups of cyclic groups,
- determine conjugacy classes, cosets and factor groups in certain groups,
- state, prove and apply Lagrange's theorem.

• Binary operations. Multiplication tables, associativity, commutativity, associative powers. [2 lectures]
• Groups. Definitions and examples (groups of numbers, the integers modulo n, symmetric groups, groups of matrices). [2]
• Subgroups. Subgroup criterion, cyclic subgroups, centralizer, centre, order of an element. [4]
• Cyclic groups. Subgroups of cyclic groups are cyclic, subgroups of finite cyclic groups. [1]
• Cosets and Lagrange's Theorem. [2]
• Homomorphisms and isomorphisms. Definition and examples, group theoretic properties. [2]
• Conjugacy. Conjugacy classes, conjugacy in symmetric groups, the class formula. [4]
• Normal subgroups. [2]
• Factor groups. [2]
• The First Isomorphism Theorem [1]

• Coursework; An in-class test, weighting within unit 20%
• End of semester examination; Weighting within unit 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

John B. Fraleigh, A First Course in Abstract Algebra, Addidon-Wesley

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.