## 5E: Topics in Algebra MATHS5077

• Academic Session: 2024-25
• School: School of Mathematics and Statistics
• Credits: 10
• Level: Level 5 (SCQF level 11)
• Typically Offered: Semester 1
• Available to Visiting Students: No
• Collaborative Online International Learning: No

### Short Description

This course builds on the concepts of groups, rings and fields. 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

### Excluded Courses

4H: Topics in Algebra (MATHS4111)

### Assessment

Assessment

90% Examination, 10% Coursework.

Reassessment

MSc students will have a resit if necessary

Main Assessment In: April/May

### Course Aims

In algebra, groups encode symmetries of many different kinds of objects. The main aim of this course is the development of advanced tools for classifying certain families of groups, as well as a depth and breadth of understanding and application of these tools. 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 Schreier and Jordan-Hölder Theorems 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 and to classify some 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

(i) identify known groups from a group presentation; perform advanced calculations with groups presentations;

(j) state and prove results about irreducible polynomials and field extensions

### 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.