## MAS114 Numbers and Groups

Both semesters, 2020/21 | 20 Credits | ||||

Lecturer: | Dr Evgeny Shinder | Home page | Timetable | Reading List | |

Aims | Outcomes | Teaching Methods | Assessment | Full Syllabus |

The module provides an introduction to more specialised Pure Mathematics. The first half of the module will consider techniques of proof, and these will be demonstrated within the study of properties of integers and real numbers. The second semester will study symmetries of objects, and develop a theory of symmetries which leads to the more abstract study of groups.

There are no prerequisites for this module.

The following modules have this module as a prerequisite:

MAS220 | Algebra |

MAS221 | Analysis |

MAS344 | Knots and Surfaces |

MAS430 | Analytic Number Theory |

## Outline syllabus

Semester One:- Sets and functions
- Logic
- The natural numbers and the principle of induction
- How to write proofs
- The integers and divisibility
- The integers (mod n) and congruences
- The real numbers and convergent sequences

- Functions, symmetries and permutations
- Groups, subgroups and cyclic groups
- Relations and modular arithmetic
- Lagrange's Theorem
- Group actions and colouring problems

## Aims

- Introduce students to the language of university mathematics;
- strengthen their ability to write mathematical (and other) arguments in a careful and logical manner;
- give students an understanding of what constitutes a proof and to foster an appreciation of the need for precision;
- motivate an appreciation of abstraction as applied to the theory of groups.

## Learning outcomes

- appreciate the importance of careful definitions, clear writing and thinking;
- be able to construct simple proofs in a number theoretic setting;
- understand different types of proofs, including induction and contradiction;
- understand the role of counterexamples and be able to supply them in suitable situations;
- solve linear diophantine equations and elementary linear congruences;
- understand properties of the real numbers;
- be able to prove basic convergence of sequences;
- test simple functions for injectivity, surjectivity and bijectivity and test relations for reflexivity, symmetry and transitivity;
- decide whether given infinite sets of numbers are countable;
- express permutations as products of disjoint cycles and of transpositions and determine the sign and parity of a permutation;
- demonstrate knowledge of basic notions of group theory;
- work with basic examples of groups;
- use the subgroup criterion to test whether subsets of familiar groups are subgroups;
- use Lagrange's Theorem to demonstrate the non-existence of subgroups of a given order.

## Teaching methods

In both semesters 1 and 2, there will be two formal lectures per week, which will involve the formulation of new theory and worked examples. Each week students will be set exercises, and they will attend one tutorial each week where their solutions to the exercises will be discussed. Areas of common difficulty will be explained on the board by the tutorial leader. Students will also submit their solutions to the exercises for feedback, although this will not be part of the formal assessment.

44 lectures, 22 tutorials

## Assessment

A formal, closed book, two hour examination at the end of the second semester (80%), and weekly online tests (20%).

## Full syllabus

Semester One:

**Introduction**

(1 lecture)

**1. Sets and functions**

(2 lectures) Some of the basic building blocks of formal mathematics. Notation for set operations. Functions, including composition, and definitions of injective, surjective, bijective.

**2. Logic**

(1 lecture) The basic principles of logic including logical implication, converse, contrapositive, negation, "and" and "or". Truth tables. Quantifiers.

**3. The natural numbers and the principle of induction**

(2 lectures) The basic formalism with examples and false proofs. Variants including strong induction. The well-ordering axiom and equivalence with the principle of induction.

**4. How to write proofs**

(1 lecture)

**5. The integers and divisibility**

(5 lectures) Divisibility, coprimality, prime and composite numbers. Euclid's Theorem. Division with remainder and Euclid's algorithm. The Fundamental Theorem of Arithmetic. Linear Diophantine equations.

**6. The integers (mod n) and congruences**

(5 lectures) Congruences and linear congruence equations. The Chinese Remainder Theorem and simultaneous congruences. Fermat's Little Theorem. Wilson's Theorem. Public Key Cryptography.

**7. The reals and convergent sequences**

(5 lectures) The rational numbers and irrational numbers. √2 is irrational. Convergent sequences and the construction of the reals from the rationals by Cauchy sequences.

Semester Two:

**1. Functions and Symmetries**

(5 lectures) [Chapters 1,2 in the core text] Functions. Examples including rotations and reflections in

**R**

^{2}. Symmetries of the square and circle; group properties. Domain and codomain of a function. Composition of functions, the associative law. The identity function on a non-empty set. Inverse functions; injective, surjective and bijective functions. A function has an inverse if and only if it is bijective. Countability.

**Q**is countable but

**R**is not.

**2. Permutations**

(4 lectures) [Chapters 2,7] Permutations, group properties. Cycles and transpositions. Algorithms to express a permutation as products of disjoint cycles/transpositions. Discussion of uniqueness in such products. Parity and sign. sgn (θ°ϕ) = (sgn θ)(sgn ϕ).

**3. Groups and Subgroups**

(4 lectures) [Chapters 4,5,8] The group axioms. Examples of groups to include groups of numbers, groups in modular arithmetic, groups of matrices and groups of functions, in particular the groups D

_{4}and O

_{2}(of symmetries) and S

_{n}(of permutations). The order of a group. Abelian groups. Consequences of the axioms: uniqueness of neutral element and inverses, Latin square property, cancellation. Direct products. Brief discussion of isomorphism and homomorphism, isomorphisms between D

_{3}and S

_{3}and between D

_{4}and a subgroup of S

_{4}. Subgroups, examples, the subgroup criterion. Klein's 4-group and the dihedral groups D

_{n}, n ≥ 3, as subgroups of O

_{2}. The alternating group.

**4. Cyclic groups**

(2 lectures) [Chapter 6] Cyclic subgroups and cyclic groups. The order of an element. Every subgroup of a cyclic group is cyclic.

**5. Group Actions**

(2 lectures) [Chapters 7,11,12] Group actions; examples. Orbits and stabilizers. A formula for the number of orbits of the action of a finite group on a finite set (proof later). Application to colouring problems.

**6. Equivalence relations and modular arithmetic**

[Chapter 9] (1-2 lectures) Relations, reflexive relations, symmetric relations, transitive relations, equivalence relations. Equivalence classes. Distinct equivalence classes are disjoint. Partitions. Congruence modulo n is an equivalence relation. Reminders on modular arithmetic.

**7. Cosets and Lagrange's Theorem**

(1-2 lectures) [Chapter 10] Left cosets. Lagrange's Theorem. Groups of prime order are cyclic. Fermat's Little Theorem.

**8. The Orbit-Stabilizer Theorem**

(1-2 lectures) [Chapter 11] Interpretation of the left coset gH of a stabilizer H=stab(x) as the set {k ∈ G:k*x=y}, where g*x=y. The Orbit-Stabilizer Theorem. Its application to prove the orbit-counting formula used earlier.

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

B |
Allenby | Numbers and proofs | 511.36 (A) | Blackwells | Amazon |

B |
C. R. Jordan and D. A. Jordan | Groups | 512.86 (J) | Blackwells | Amazon |

B |
Mason | Learning and doing mathematics | 510 (M) | Blackwells | Amazon |

B |
Mason | Thinking mathematically | 510 (M) | Blackwells | Amazon |

C |
Burn | A Pathway into Number Theory | 512.81(B) | Blackwells | Amazon |

C |
Cupillari | The nuts and bolts of proofs | 510.1 (C) | Blackwells | Amazon |

C |
Eccles | An introduction to mathematical reasoning | 510.1 (E) | Blackwells | Amazon |

C |
Polya | How to solve it | 510 (P) | Blackwells | Amazon |

C |
Solow | How to read and do proofs | 511.36 (S) | Blackwells | Amazon |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.