Mathematics: Infinite Groups

The course provides a rigorous introduction to infinite groups, focusing on their construction, properties, and role in modern mathematics. By the end of the course, you should be able to explain in detail the concepts, theorems, and methods introduced, identify the most important results, and present their proofs. You will also learn to integrate knowledge from different parts of the course when solving problems, and to communicate solutions clearly and coherently in both speech and writing.

Topics include

• Finitely generated groups: free groups, Nielsen-Schreier Theorem, group presentations, finitely presented groups.
• Free products: free products with amalgamation, HNN extensions.
• Solvable groups: polycyclic groups, nilpotent groups.
• Subgroups: finite-index subgroups, virtual properties, maximal subgroups.
• Residually finite groups: Hopfian groups, Malcev's Theorem, Baumslag-Solitar groups.
• The Generalised Burnside Problem: variants of the Burnside Problem and groups acting on rooted trees.

The course is alternative-compulsory within the Master’s programme in Mathematics at Lund University. It can also be taken as an optional course within the Bachelor’s programme in Mathematics, or as a standalone course.

Teaching combines lectures and seminars, where theoretical exposition is integrated with discussion and problemsolving. Students are expected to work actively with exercises, independently identify problems that can be solved using methods from the course, and present solutions orally and in writing with correct terminology. Assessment consists of a written exam, followed by an oral exam for those who pass the written part, and includes opportunities to retake the written exam if needed. Adapted examination formats are available in consultation with Disability Support Services.

This course prepares you for advanced studies in algebra and group theory, and strengthens your foundation for research or analytical roles in academia and industry. You will develop problemsolving skills, abstract reasoning, and the ability to evaluate solution strategies critically. By completing the course, you will also be able to argue for the importance of infinite groups as a significant part of group theory, and appreciate their role in broader areas of mathematics.

Prerequisites

For admission to the course, English 6 / B is required as well as at least 90 credits in mathematics including the course MATC31 Algebraic Structures, 7.5 credits or equivalent.

Tuition fees for non-EU/EEA citizens

Citizens of countries outside:

• The European Union (EU)
• The European Economic Area (EEA) and
• Switzerland

are required to pay tuition fees. You pay an instalment of the tuition fee in advance of each
semester.

Tuition fees, payments and exemptions

Full programme/course tuition fee: SEK 21,250
First payment: SEK 21,250

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Note that you may also need to pay an application fee, or provide proof of exemption.

Application fee

No tuition fees for citizens of the EU, EEA and Switzerland

There are no tuition fees for citizens of the European Union (EU), the European Economic Area (EEA) and Switzerland.