Course Catalogue

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This course builds on the concepts of groups 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.

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

Mandatory Entry Requirements

3H Algebra (MATHS 4072)

Recommended Entry Requirements

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.

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. 

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.

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

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

(b) state and prove the Sylow Theorems; 

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

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

(e) use elementary divisors and invariant factors to determine isomorphism classes of finitely generated abelian groups;

(f) compute the Normal Form of a finitely generated abelian group;

(g) state and apply the Universal Property of free groups;

(h) manipulate group presentations to compute basic invariants of the group.

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