Introduction to Proofs and Group Theory
This unit aims to to introduce students to fundamental concepts in Mathematics including set theory, techniques of proof and group theory.
The first half provides an introduction to logical propositions, basic set theory and cardinality, functions and relations, and proof techniques. These notions are exemplified with some topics from elementary number theory, such as the Fundamental Theorem of Arithmetic, Euclid’s algorithm, modular arithmetic.
The second half explores the area of group theory. In the past, certain systems studied in various parts of mathematics have turned out to have common features, and these have been formalised into the definition of a group. Some of the earliest examples arose in connection with the solution of polynomial equations by formulae, and involved what we would now call groups of permutations. Other examples arise in trying to pin down mathematically what it means to say that a geometrical figure is symmetric and to quantify just how symmetric it is. It makes sense to study in one go all the systems which have the same general features. We shall start from the formal definition of a group and derive important general results from it using careful mathematical reasoning, but throughout there will be an emphasis on particular examples in which calculations can be performed relatively easily. The unit aims to introduce students to basic material in group theory, including examples of groups, group homomorphisms, subgroups, quotient groups, basic theorems on groups (such as Lagrange’s Theorem, Fermat’s Little theorem, 1st Isomorphism Theorem) and their applications.
At the end of the unit, the students should:
- be able to distinguish correct from incorrect and sloppy mathematical reasoning,
- be able to understand and write clear mathematical statements and proofs;
- be able to produce proofs using mathematical induction;
- be able to correctly use quantifiers and to negate logical statements that include quantifiers and connectives;
- be able to work with sets, whether they are finite, countable, or uncountable;
- be proficient in establishing fundamental properties of functions, such as injectivity and surjectivity;
- be proficient in using Euclid’s algorithm and modular arithmetic.
- be able to demonstrate facility in working with various specific examples of groups;
- be able to solve standard types of problems in introductory group theory;
- understand and be able to apply the basic concepts and results presented throughout the unit.
Reading and References
- Larry Gerstein, Introduction to Mathematical Structures and Proofs, Springer, 2008.
- D.J. Velleman, How to Prove It: A Structured Approach, Cambridge University Press, 2006.
- P.J. Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, Cambridge University Press, 1997.
- C. R. Jordan and D. A. Jordan, Groups, originally published by Edward Arnold in 1994, reprinted by Newnes (Elsevier) in 2001, 2003, 2004; ISBN 034061045x.
An A in A level Mathematics or equivalent.
Methods of teaching
Lectures, supported by lecture notes with problem sets and model solutions, problems classes and small group tutorials.
Methods of Assessment
The pass mark for this unit is 40.
- Problem sheets set by the lecturer and marked by the students’ tutors.
- Two 1.5h exams (45% each)
- Coursework (10%)
For information resit arrangements, please see the re-sit page on the intranet.
Further exam information can be found on the Maths Intranet.