# Semester 1 2022-23

## Welcome

This module is an introduction to Group Theory for students studying Mathematics at NUI Galway in the College of Science & Engineering or the College of Arts. This website will be the main online resource for the module. The course Blackboard page will be used for announcements and submissions of assignments, and the gradebook there will be used for keeping a record of continuous assessment. The lecture notes form the “text” for the course. As a backup, links to the video lectures that were used in the 2020/21 academic year, when the module ran entirely online, are included wit the week-by-week online content.

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 This is out of date but the information on the curriculum content is still relevant – please ignore details about assessment, which are not current.

## Lectures

Thursday 12.00–12.45, AC202 and
Friday 12:00–12:45, AC201

Lectures will be held in person, and not streamed or recorded. As a backup to the in-person lectures, video lectures from the 2020-21 academic year (when all engagement was remote) will be made available at the weekly pages. If you are using the video lectures, please ignore any comments there about assessment, as they do not refer to this year’s instance of the module. The curriculum has not changed though, so the lecture content in the videos is still relevant.

## Tutorials

Tutorials commence in Week 4, with times to be announced. 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).

## Syllabus Outline

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.

## Course and Lecture Notes

The lecture notes (updated on October 18) form the main text for the course. This is the complete set of lecture notes (which will also be posted in instalments on the weekly pages). It will probably be updated as the semester proceeds, so please download it again in a few weeks (this version is from September 6).

## Assessment

Assessment of students’ learning in Groups MA3343 will consist of the following elements.

• Two homework assignments
Distributed in Week 2 or 3 and in Week 6, via Blackboard. Submission dates in Week 6 and Week 9. More advice later about homework assignments. (12.5% each)
• 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. We’ll have a poster exhibition in Week 12. (25%)
• 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.