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
Having successfully completed this module you will be able to:
- Derive the existence of groups of a specified small order
- Recall and use the definitions and properties of dihedral, symmetric and alternating groups
- Recall and use the definitions and properties of cossets and subgroups
- Verify group properties in particular examples
- Understand, use the properties of and manipulate permutations
- Recall and apply Lagrange's theorem
- Understand and use the terms homomorphism and isomorphism
- Understand and use the properties of group actions
- Understand and use the concept of conjugacy
- Recall and apply Burnside's formula
- Recall and apply Sylow’s Theorems to determine the structure of certain groups of small order
- 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
|Total study time||150|
Resources & Reading list
Rotman, J J. A first module in Abstract Algebra with Applications. Pearson.
Fraleigh John B. A First Module in Abstract Algebra. Addison Wesley.
Armstrong MA. Groups & Symmetry. Springer.
Robinson, D J S. An Introduction to Abstract Algebra. de Gruyter.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External