This module is an introduction to Group Theory for students studying Mathematics at NUI Galway in the Colleges of Science and Arts. This website will be the main online resource for the module. The course Blackboard page will be used for announcements and the gradebook there will be used for keeping a record of continuous assessment. Each week will have its own page, with a summary of the week's content. The lecture notes form the "text" for the course and they are posted on this front page, and also link ed from the pages for each week. Also included in the weekly pages are links to the video lectures that were used in the 20/21 academic year, when the module ran entirely online.
Here is a welcome video (from 20/21) with some introductory information. This is out of date but the information on the curriculum content is still relevant - please ignore details about assessment, which are not current.
Thursday 12.00--12.45, AC202 and
Friday 12:00--12:45, AC201
Lectures will be held in person. We will protect ourselves and each other by keeping windows open and by wearing face coverings. For everyone's safety, both students and lecturer will wear face coverings at all times during lectures. Any student who is exempt from wearing one is asked to inform the lecturer of this by email as soon as possible, and in advance of the first lecture on September 9. Any student who is unable or reluctant to attend lectures in person, because of the safety risk, is also invited to make early contact with the lecturer, who will be happy to discuss the possibility of remote participation.
Tutorials
Tutorials commence in Week 4, on Wednesdays at 2.00 and Thursdays at 1.00.
The tutor is Koushik Paul, who is a PhD researcher in group theory. The Wednesday tutorial will be held online via Zoom, and the Thursday tutorial in person on campus (venue to follow).
Content will be added here Week by Week. Each week's content will include a summary of the relevant topic, last year's video lectures (which are included as a backup to this year's lectures on campus), and links to the relevant section of two of the lecture notes. The lecture notes constitute the text for the course and generally have more (and different) content than the lectures.
This course is an introduction to Group Theory. The subject is concerned with algebraic structures (sets) whose elements can be combined in pairs according to some operation (such as addition, multiplication, composition) in a manner that satisfies certain natural conditions known as the axioms of a group. Groups are defined in abstract terms but they are ubiquitous in mathematics, examples include the integers (with addition), the non-zero rational numbers (with multiplication), the permutations of a set (with composition), the 2 by 2 real matrices of non-zero determinant (with matrix multiplication), and so on. The idea of using the single word ``group'' to describe a diverse range of objects that share some properties in terms of their algebraic structure is a relatively newfangled one, but it has proved to be extremely powerful. Although abstract reasoning is a central and essential feature of group theory, we will have plenty of examples to give us some context. Many of our concrete examples will come from matrix algebra, which provides both a plentiful supply of interesting examples and a concrete structure for representing groups that arise in different situations.
Our syllabus will have four chapters.
What is a Group?
Examples of groups, axioms of a group, subgroups and generating sets.
Essential concepts of group theory
Abelian groups, the centre of a group, the centralizer and conjugacy class of an element. The order of a group and the order of an element. Lagrange's Theorem on the order of a subgroup of a finite group.
Group actions
Groups acting on sets, with examples. The Orbit-Stabilizer Theorem. Isomorphism. Cayley's Theorem.
Quotient Groups
Group homomorphisms. Normal subgroups and quotient groups.
Assessment of students' learning in Groups MA3343 will consist of the following elements.
One weekly challenge
Each week (except Week 12) will include a challenge, which involves preparing one page (maybe in the form of a slide) that can be shared with the class. It could be a picture, or an example, or an explanation, or a proof. Up to 2% for each successfully completed challenge (up to 20% in all). More details on this in the first week's lectures. Note on marking of weekly challenges: Identical submissions will get zero or very low marks. Working together is encouraged, but make a joint
submission (up to three named authors) in that case. Submissions
transcribed verbatim from elsewhere will get zero or very low
marks. The weekly challenges are about your own creative
expression - searching for relevant texts is encouraged, but giving
your own explanation is part of the task.
Two homework assignments
Distributed in Week 2 or 3 and in Week 6. Submission dates in Week 6 and Week 9. More advice later about homework assignmnents. (12.5% each) The first homework assignment is
here, posted on September 24 with a due date of October 15. Submission is by .pdf upload to Blackboard, see the ``Assignements" section of the MA3343 Blackboard page. When working on this homework sheet, please think about communicating your answers as clearly as completely as possible to a reader, as well as about solving the problem to your own satisfaction.
Poster project
In teams of 2 or 3, students will create a poster on a topic in Group Theory (advice and guidance will be provided, both on researching potential topics and on producing a poster). You can see more detail on the poster project
here. We'll have a poster exhibition in Week 12. You can see the posters from the 2019 exhibition here. 25% for the poster project.
Final exam
Final exam in the Semester 1 exam session in December. Up to 70%.
As you can see, this adds up to more than 100%. You are not obliged to complete all elements of the assessment, and you are encouraged to give particular attention to those parts that appeal to you the most.
Please inform yourself about the NUI Galway policy on plagiarism. If you collaborate with other students in the class on a homework problem and knowingly submit very similar solutions, please make a note of this in your submitted work. If you submit a solution that relies heavily on a book or online resource, note this also and cite the source. Do not solicit solutions from online services.
You are more than welcome to contact the lecturer about any difficulties that you encounter with the assessment or any aspect of the module.
Outline lecture notes will be posted here in instalments as the module proceeds. They are also linked weekly from the content section. For the moment they are still here from last year. They will be updated for this year as the course proceeds. There won't be too many changes from last year, but do please check for updates from time to time - they will be noted on the webpages.