# BSc Computer Science and Mathematics / Course details

Year of entry: 2020

## Course unit details:Group Theory

Unit code MATH32001 10 Level 3 Semester 1 Department of Mathematics No

### Overview

The ideal aim of Group Theory is the classification of all groups (up to isomorphism). It will be shown that this goal can be achieved for finitely generated abelian groups. In general, however, there is no hope of a similar result as the situation is far too complex, even for finite groups. Still, since groups are of great importance for the whole of mathematics, there is a highly developed theory of outstanding beauty. It takes just three simple axioms to define a group, and it is fascinating how much can be deduced from so little. The course is devoted to some of the basic concepts and results of Group Theory.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Algebraic Structures 2 MATH20212 Pre-Requisite Compulsory

Students are not permitted to take more than one of MATH32001 or MATH42001 for credit in the same or different undergraduate year.  Students are not permitted to take MATH42001 and MATH62001 for credit in an undergraduate programme and then a postgraduate programme.

### Aims

This lecture course unit aims to introduce students to some more sophisticated concepts and results of group theory as an essential part of general mathematical culture and as a basis for further study of more advanced mathematics.

### Learning outcomes

On successful completion of this course unit students will be able to:

• use the subgroup criterion to prove that various subsets are subgroups of some given group,
• calculate in small degree symmetric groups and 2x2 matrices such things as centralizers, normalizers and conjugacy classes. To be able to state the classification theorem for finitely generated abelian groups, be able to calculate torsion coefficients and determine whether two abelian groups are isomorphic.
• work with G-sets and calculate G-orbits,
• to be able to state the definition of a simple group, calculate composition factors and composition series of certain groups and be able to prove the Jordan Holder theorem,
• to be able to prove the existence of Sylow p-subgroups in finite groups. To be able to state and apply Sylow's theorems to prove certain groups cannot be simple,
• determine Sylow p-subgroups and their related properties in small order groups.

### Syllabus

1. Revision of basic notions (subgroups and factor groups, homomorphisms and isomorphisms), generating sets, commutator subgroups. [2 lectures]
2. Abelian groups, the Fundamental Theorem on finitely generated abelian groups. 
3. The Isomorphism Theorems. 
4. Simple groups, the simplicity of the alternating groups. 
5. Composition series, the Jordan-Hölder Theorem. 
6. Group actions on sets, orbits, stabilizers, the number of elements in an orbit, Burnside's formula for the number of orbits, conjugation actions, centralizers and normalizers. 
7. Sylow's Theorems, groups of order pq, pqr. 

For MATH42001 the lectures will be enhanced by additional reading on related topics.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework: in-class test weighting 20%
• End of semester examination: two hours, weighting 80%

### Feedback methods

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.

Recommended text:

John B Fraleigh, A First Course in Abstract Algebra, (5th edition), 1967, Addison-Wesley.

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 67

### Teaching staff

Staff member Role
Peter Rowley Unit coordinator