Group Theory | MATH2003 | University of Southampton

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.

Linked modules

Prerequisite: MATH1049

Aims and Objectives

Learning Outcomes

Having successfully completed this module you will be able to:

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 properties of group actions
Understand and use the terms homomorphism and isomorphism
Understand and use the concept of conjugacy
Recall and apply Lagrange's theorem
Verify group properties in particular examples
Recall and use the definitions and properties of cossets and subgroups
Recall and use the definitions and properties of dihedral, symmetric and alternating groups
Derive the existence of groups of a specified small order
Understand, use the properties of and manipulate permutations

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

Study time
Type	Hours
Teaching	48
Independent Study	102
Total study time	150

Resources & Reading list

Textbooks

Armstrong MA. Groups & Symmetry. Springer.

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

Fraleigh John B. A First Module in Abstract Algebra. Addison Wesley.

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

Assessment

Summative

This is how we’ll formally assess what you have learned in this module.

Breakdown
Method	Percentage contribution
Coursework	20%
Written exam	80%

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Breakdown
Method	Percentage contribution
Exam	100%

Repeat Information

Repeat type: Internal & External