## MATH20201 - Algebraic Structures 1

Year: 2 - Semester: 1 - Credit Rating: 10

#### Aims

The course 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.

#### Brief Description

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.

#### Learning Outcomes

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.

#### Syllabus

• 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]

#### Teaching & Learning Process (Hours Allocated To)

Lectures Tutorials/Example Classes Practical Work/Laboratory Private Study Total
33 0 0 67 100

#### Assessment and Feedback

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