The University of Southampton
Courses

# MATH2003 Group Theory

## Module Overview

Group theory is one of the great simplifying and unifying ideas in modern mathematics. It was introduced in order to understand the solutions to polynomial equations, but only in the last one hundred years has its full significance, as a mathematical formulation of symmetry, been understood. It plays a role in our understanding of fundamental particles, the structure of crystal lattices and the geometry of molecules. In this module we build on material from Linear Algebra II and will begin by revising the simple axioms satisfied by groups and begin to develop basic group theory by reference to some elementary examples. We will analyse the structure of 'small' finite groups, and examine examples arising as groups of permutations of a set, symmetries of regular polygons and regular solids, and groups of matrices. We will develop the notions of homomorphism, normal subgroups and quotient groups and study the First Isomorphism Theorem and its application. We will also examine how the notion of a permutation group can be generalized to that of a group action on a set, and will show how to use this in certain counting problems arising in combinatorics. We will also see how to use group actions to prove strong results about the structure of finite groups. We shall study Sylow’s Theorems and some of their applications.

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Verify group properties in particular examples
• Recall and apply Burnside's formula
• Recall and apply Sylow’s Theorems to determine the structure of certain groups of small order
• Understand and use the terms homomorphism and isomorphism
• Understand and use the concept of conjugacy
• Derive the existence of groups of a specified small order
• Recall and use the definitions and properties of cossets and subgroups
• Recall and apply Lagrange's theorem
• Understand, use the properties of and manipulate permutations
• Recall and use the definitions and properties of dihedral, symmetric and alternating groups
• Understand and use the properties of group actions

### Syllabus

• Revision form Linear Algebra II covering topics such as: • Matrix and disjoint cycle notation for permutations, expressing elements as products of transpositions, definitions and basic lemmas of abstract group theory: the uniqueness of the identity element and inverses • Permutation groups ; Matrix groups G1(n), S1(n), O(n), SO(n) ; The group of intergers mod n. • Cayley tables of small Symmetric Groups, Groups of Symmetries, dihedral groups (of small order) • Powers of elements • Order and generators for groups, orders of elements – examples Isomorphism and examples • Cyclic groups and their classification. • Any symmetric group is generated by transpositions • Odd and even permutations and the Alternating group • Conjugacy and conjugacy classes; example: cycle structures in symmetric groups and the relationship with conjugacy classes. • Definition and examples of subgroups and Lagrange's theorem; examples including cyclic groups and the Symmetric Groups • Testing for a subgroup. Intersections of subgroups are subgroups • Isomorphisms, their properties and the classification of small groups • Direct products and the internal direct product theorem • Normal subgroups, centre of a group, Cosets and quotient groups • Homomorphisms, kernel and image;the First Isomorphism Theorem • Group actions and Cayley’s theorem • Orbit stabiliser theorem and applications; the action of the dihedral groups on geometric features of regular n-gons, the rotation group of a cube is isomorphic to the symmetric group on 4 letters. • Burnside’s formula and its applications to enumeration of orbits e.g, counting seating plans, necklaces coloured dice • Application of group actions to finite group theory: Class equation, finite p-groups, Sylow’s theorems and their applications.

### Learning and Teaching

#### Teaching and learning methods

-

TypeHours
Independent Study102
Teaching48
Total study time150

Fraleigh John B. A First Module in Abstract Algebra.

Armstrong MA. Groups & Symmetry.

Rotman, J J. A first module in Abstract Algebra with Applications.

Robinson, D J S. An Introduction to Abstract Algebra.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 40%
Written assessment 60%

#### Referral

MethodPercentage contribution
Written assessment 100%

#### Repeat Information

Repeat type: Internal & External

Prerequisite: MATH1049

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.