Group Theory

Course Objective

* The student knows Z/nZ, (Z/nZ)^*, the Chinese remainder theorem, and
the 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, stabiliser, 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 binary
operation 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.

Additional Information Teaching Methods

Lectures and tutorials, both two hours per week during 14 weeks, and
study 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 assignments
to 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.

Additional Information Target Audience

BSc Mathematics Year 1

Recommended background knowledge

The VU course Basic Concepts in Mathematics, as well as some parts of
the VU courses Linear Algebra (in particular, matrices) and Discrete
Mathematics (in particular, the part on permutation groups).