# 2007/08 Undergraduate Module Catalogue

## MATH1022 Introductory Group Theory

### 10 creditsClass Size: 250

**Module manager:** Dr R. Marsh**Email:** marsh@maths.leeds.ac.uk

**Taught:** Semester 2 View Timetable

**Year running** 2007/08

### Pre-requisite qualifications

The pre-requisites for this module are either (MATH1015 & MATH1035) or (MATH1050 & MATH1060).**This module is approved as an Elective**

### Module summary

Group theory may be regarded as an abstract study of symmetry. Thus for a typical geometrical figure, its degree of symmetry may be captured by the corresponding group, certainly how many symmetries there are, but also, precisely how they interact (the `structure' of the group). Groups play a central role in mathematics and its applications. This course treats the basic theory as far as Lagrange's theorem (the order of a subgroup divides the order of the group) and quotient groups.### Objectives

To give a coherent treatment of the fundamental ideas of group theory. To illustrate how the groups corresponding to various geometrical shapes and patterns correspond to their degree and type of symmetry. On completion of this module, students should be able to: a) determine whether or not a given structure is a group; b) describe groups of rotations and isometries, and to identify their subgroups; c) perform computations in finite cyclic groups, and relate this to calculations involving congruences; d) calculate using permutations, and determine the order of a permutation; e) list the families of cosets of various groups of small order.### Syllabus

Definitions and examples of groups. Basic terminology. Symmetries of geometrical figures, and isometries. Multiplicative group of units of Z / n Z. Additive groups. Subgroups. Subgroup criterion. Examples. Order of an element. Powers of elements of finite order. Cyclic groups. Direct product of two groups. Product of two cyclic groups is cyclic if and only if their orders are coprime. Isomorphisms and homomorphisms. Cosets and Lagrange's theorem. Order of an element of a finite group divides the order of the group. Groups of order p, 4. Intersections of subgroups. Subgroups of the dihedral groups of order 6 and 8. Fermat's little theorem. Permutations. Cycle notation. Order of a permutation. Even and odd permutations. Sn and An. Conjugacy, centre, normal subgroups. Centralizer of an element. Number of conjugates = index of centralizer. Conjugacy classes in Sn. Quotient groups and the first isomorphism theorem.

### Teaching methods

Delivery type | Number | Length hours | Student hours |

Lecture | 22 | 1.00 | 22.00 |

Tutorial | 11 | 1.00 | 11.00 |

Private study hours | 67.00 | ||

Total Contact hours | 33.00 | ||

Total hours (100hr per 10 credits) | 100.00 |

### Methods of assessment

**Coursework**

Assessment type | Notes | % of formal assessment |

In-course Assessment | . | 15.00 |

Total percentage (Assessment Coursework) | 15.00 |

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

**Exams**

Exam type | Exam duration | % of formal assessment |

Standard exam (closed essays, MCQs etc) | 2 hr 00 mins | 85.00 |

Total percentage (Assessment Exams) | 85.00 |

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

### Reading list

The reading list is available from the Library websiteLast updated: 18/03/2008

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