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 - Foundations of Pure Mathematics A (Compulsory)
- MATH10111 - Foundations of Pure Mathematics 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:
- To be able to state the group axioms and be familiar with frequently met examples of groups
- Give the definition of a subgroup is and be able to employ the subgroup criterion to determine whether certain subsets of a group are subgroups .
- Be able to work out conjugacy classes in certain groups.
- Give the definition of a cyclic group, be familiar with examples of such and their important properties such as their generators and subgroups.
- To be able to give the definition of cosets and their fundamental properties and to calculate in small examples.
- To be able to give the construction of factor groups and be able to calculate them in small examples.
- To be able to the state Lagrange's theorem and give its proof, as well as some elementary applications of this result.
- 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
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.
- Lectures - 33 hours
- Independent study hours - 67 hours