### General Information

Course Code | X_401105 |
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Credits | 6 EC |

Period | P4+5 |

Course Level | 200 |

Language of Tuition | English |

Faculty | Faculty of Science |

Course Coordinator | prof. dr. R.M.H. de Jeu |

Examiner | prof. dr. R.M.H. de Jeu |

Teaching Staff |
prof. dr. R.M.H. de Jeu |

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods | Seminar, Lecture |
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Target audiences

This course is also available as:

### Course Objective

* The student knows Z/nZ, (Z/nZ)^*, the Chinese remainder theorem, andthe Euler phi-function, and can solve problems about and with these.

* The student knows basic concepts from group theory (including

subgroups, cyclic groups, generators, dihedral groups, permutation

groups, centre, commutator subgroup, normal subgroup, coset,

homomorphism, quotient group, group action, stabilisor, orbit) and can

solve problems about and with those in explicit situations.

* The student knows basic theorems from group theory (including the

theorems of Cauchy, Lagrange, and the first isomorphism theorem) and can

use these to compute and/or prove certain properties in explicit

situations.

### Course Content

We study an algebraic structure (called a group) with one binaryoperation that satisfies certain properties. Examples of such groups are

the integers or real numbers under addition, invertible matrices (of a

fixed size) under matrix multiplication, or bijections from a given set

to itself under composition of functions. Groups also show up in many

situations as the symmetries of an object or structure. By formalising

their common properties we can prove various general results about

groups, which we also illustrate by working out what they mean in

various concrete cases.

We treat the following topics.

* The integers modulo n; Chinese remainder theorem, Euler phi-function.

* Abstract definition of a group, order of a group or element of a

group.

* Examples of groups (the integers, the integers modulo n, dihedral

groups, matrix groups, etc.).

* Subgroups, generators, homomorphisms.

* Normal subgroups and quotient groups.

* Cosets, index of a subgroup, Theorem of Lagrange.

* The first isomorphism theorem.

* Commutator subgroup, homomorphism theorem.

* Group actions, orbits, the class equation, Burnside's lemma.

* Theorem of Cauchy.

### Teaching Methods

Lectures and tutorials, both two hours per week during 15 weeks, andstudy sessions, two hours every other week. Students also make a number

of computer-supported assignments, and must hand in a written assignment

every other week.

### Method of Assessment

For this course there will be two partial exams, six written assignmentsto be handed in (out of which the best five count towards the grade),

some computer-supported assignments, as well as a resit. The grade is

determined as follows:

A. Based on the two partial exams: the written assignments count for 5%

in total, the computer-supported assignments count for 5% in total, the

average score for the two partial exams counts for 90%;

B. Based on the resit: the written assignments count for 5% in total,

the computer-supported assignments count for 5% in total, the score of

the resit counts for 90%; or, if this results in a higher grade, the

score of the resit counts for 100%.

NB: In order to take part in the partial exams, you must have been

present at at least 70% of the study sessions and tutorials (full time

students only). There is no such attendance requirement in order to take

part in the resit.

### Literature

David S. Dummit, Richard M. Foote, "Abstract algebra", 3rd edition(2003), John Wiley and Sons.

### Target Audience

BSc Mathematics Year 1### Recommended background knowledge

The VU course Basic Concepts in Mathematics, as well as some parts ofthe VU courses Linear Algebra (in particular, matrices) and Discrete

Mathematics (in particular, the part on permutation groups).