Algebraic Structures 1
|Unit level:||Level 2|
|Teaching period(s):||Semester 1|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
- MATH10101 - Sets, Numbers and Functions A (Compulsory)
- MATH10111 - Sets, Numbers and Functions B (Compulsory)
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.
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.
On completion of this unit successful students will be able to:
- Appreciate and use the basic definitions and properties of groups;
- Command a good understanding of the basic properties for a good range of examples;
- Understand and find simple proofs of results in group theory.
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Coursework; An in-class test, weighting within unit 20%
- 2 hours end of semester examination; Weighting within unit 80%
- 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). 
- Subgroups. Subgroup criterion, cyclic subgroups, centralizer, centre, order of an element. 
- Cyclic groups. Subgroups of cyclic groups are cyclic, subgroups of finite cyclic groups. 
- Cosets and Lagrange's Theorem. 
- Homomorphisms and isomorphisms. Definition and examples, group theoretic properties. 
- Conjugacy. Conjugacy classes, conjugacy in symmetric groups, the class formula. 
- Normal subgroups. 
- Factor groups. 
- The First Isomorphism Theorem 
John B. Fraleigh, A First Course in Abstract Algebra, Addidon-Wesley
Tutorials will provide a place for student worked examples to be marked and discussed providing feedback on performance and understanding.
- Lectures - 33 hours
- Independent study hours - 67 hours