4H: Topics in Algebra MATHS4111

  • Academic Session: 2018-19
  • School: School of Mathematics and Statistics
  • Credits: 10
  • Level: Level 4 (SCQF level 10)
  • Typically Offered: Semester 1
  • Available to Visiting Students: Yes
  • Available to Erasmus Students: Yes

Short Description

This course builds on the concepts of groups, rings and fields covered in the 3H Algebra course. A primary focus will be on building tools for classifying certain families of groups, with one highlight being the Sylow Theorems for finite groups.

Timetable

17 x 1 hr lectures and 6 x 1 hr tutorials in a semester

Requirements of Entry

Mandatory Entry Requirements

3H Algebra (MATHS 4072)

 

Recommended Entry Requirements

Assessment

Assessment

90% Exam, 10% Coursework.

 

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? Not applicable

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

Course Aims

In algebra, groups encode symmetries of many different kinds of objects. The main aim of this course is the development of tools for classifying certain families of groups. Topics will include a closer look at group homomorphisms and isomorphisms, certain series of subgroups associated to a given group, the Sylow Theorems (which give insight into the structure of finite groups), free groups, and free abelian groups. As time permits, there may be a brief treatment of topics in rings and fields, laying some groundwork for the study of roots of polynomials.

Intended Learning Outcomes of Course

By the end of this course students will be able to:

(a) state, prove, and apply the Isomorphism Theorems for Groups;

(b) state and prove the Jordan-Hölder Theorem for composition series of groups, and calculate various composition series for specific groups;

(c) state and prove the Sylow Theorems; 

(d) apply the Sylow Theorems to analyse subgroup structure of finite groups;

(e) state and prove the Classification Theorem of Finitely Generated Abelian Groups;

(f) find normal forms of finitely generated abelian groups;

(g) compute the rank of a free group;

(h) describe all homomorphisms of free groups.

Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.