Algebra 2: Theory and Structure

This module runs in alternate years: 2013-14, 2015-16 and so on.

EMMS094S6 (30 credits)

Aims

• In this course you will encounter algebraic structures such as groups, rings, fields and vector spaces. The importance and usefulness of the axiomatic approach will be stressed.
• Groups will be studied in the most depth, and you will see some of the fundamental results and their consequences Theorem. You will become familiar with many different examples of groups, rings, fields and vector spaces, which are ubiquitous in pure mathematics.
• Applications of some of the concepts introduced will be given, for example error detecting and error-correcting codes (such as barcodes and ISBNs).

Teaching and Assessment

Teaching for this module will take place throughout the year, with eight evenings of lectures in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term.

Of the final course mark, 80% is based on a three-hour exam in June and the other 20% is from assessed coursework.

Coursework will consist of short, problem based assignments. You will have around three weeks to complete each one.

The examination in June has two sections. Section A (worth 40%) consists of compulsory short questions. Section B (also worth 40%) contains several longer questions of which you must answer two.

Syllabus

1. Group Theory: revision of group axioms, properties and basic results. Further examples of groups. Structure of groups of order p and 2p for p prime. Cosets, normal subgroups, homomorphisms, isomorphisms, correspondence between normal subgroups and kernels of homomorphisms. (4 weeks/ 12 hrs)

2. Group Actions: definition of G-sets; orbits and stabilisers; the Orbit-Stabiliser Theorem; applications to conjugation of elements and subgroups; the Orbit-counting Lemma; applications to colouring problems. (4 weeks/ 12 hrs)

3. Rings, Fields and Vector Spaces: Definition of a ring; examples; subrings and ideals; zero divisors and units; factorisation; integral domains and Euclidean domains; fields; the field of fractions; characteristic of a field; vector spaces and their properties; dimension of a vector space, bases and results; application to error-correcting and error detecting codes (8 weeks/ 12 hrs)

In addition there will be 2 weeks/6 hours of consolidation and revision lectures in the Summer Term.

Learning Outcomes

Subject Specific

1. Knowledge and understanding of, and the ability to use, mathematical and/or statistical methods and techniques.
In particular:

• the use of axioms to define concepts such as groups, rings, fields and vector spaces.
• Use of group theoretic methods to understand and analyse problems involving symmetry.

2. Knowledge and understanding of a range of results in mathematics and/or statistics
In particular:

• Knowledge of proof of elementary properties of groups.
• Familiarity with the main examples of groups, such as number sets, integers modulo n, symmetry groups and groups of permutations.
• Ability to apply results such as the subgroup criterion in specific instances.
• Awareness of the axioms for rings, fields and vector spaces.
• Familiarity with common examples of rings, fields and vector spaces.

3. Appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments.

4. Understand the importance of assumptions and an awareness of where they are used and the possible consequences of their violation.
In particular:

• the understanding that the properties of algebraic structures differ depending on the axioms used to define them.

5. Appreciation of the power of generalization and abstraction in the development of mathematical theories.

6. A deeper knowledge of some particular areas of mathematics and/or statistics.
In particular:

• the theory of groups, rings, fields and vector spaces.

Intellectual

1. Ability to comprehend conceptual and abstract material.

2. Develop a logical and systematic approach to problem solving.

Practical

1. Problem-solving skills, including the ability to assess problems logically and approach them analytically.

2. Highly developed quantitative skills.

3. Ability to transfer knowledge and expertise from one context to another.
In particular:

• the re-appearance of the group structure in the context of rings, fields and vector spaces.
• The use of the orbit counting lemma in many different contexts.
• Application of vector spaces to error detecting and error correcting codes.
• Understanding of the value of proving results for general structures such as groups, which can then be applied to numerous seemingly disparate situations.

Personal and Social

1. Ability to work independently with patience and persistence.

2. Time-management skills and organizational skills.

3. Good communication skills, including the ability to write coherently.

4. Ability to complete work in a limited time period.

Recommended Texts

• Numbers, groups and codes, J.F. Humphreys, M.Y. Prest, Cambridge University Press (1989).
• Introduction to Group Theory (Second Edition), W. Ledermann, A.J. Weir, Addison Wesley Longman, (1996)
• A Course in Group Theory, J. F. Humphreys, Oxford University Press (1996).
• Introduction to Algebra, Peter J. Cameron, Oxford University Press (1998).
• Linear Algebra, R.B.J.T. Allenby, Butterworth-Heinemann (1995)