A group is a mathematical entity with a simple structure: a set with a binary operation satisfying certain axioms. Groups are ubiquitous throughout mathematics, helping us describe symmetries in nature. Some examples of groups are integers under the operation of addition, permutations under the operation of composition, and invertible square matrices under the operation of matrix multiplication. Some real-world examples in which groups appear are symmetries of crystals, wallpaper patterns, Islamic geometric patterns, error detecting codes such as ISBN numbers on books, and the geometry of the spacetime we inhabit.

Some of the basic definitions and concepts in group theory were introduced in the Year One module MA1005 Foundations of Mathematics I. This Year Two module will build upon these concepts in order to provide an in-depth study of group theory and its most fundamental results.

• State and apply the group axioms.
• Understand the concepts of abelian groups, subgroups, generators, homomorphisms, isomorphisms, kernels, images, conjugacy, cosets, normal subgroups and quotient groups.
• Prove fundamental group theorems, including Lagrange’s theorem, the first isomorphism theorem and Cayley’s theorem.
• Work with certain examples of groups, such as cyclic, dihedral and symmetric groups.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• On-line resources that you work through at your own pace (e.g. videos, web resources, e-books, quizzes),
• On-line interactive sessions to work with other students and staff (e.g. discussions, live streaming of presentations, live-coding, group meetings)
• Face to face small group sessions (e.g. tutorials, exercise classes, feedback sessions)

Skills:
Appreciation of the general prevalence and significance of groups in mathematics. Ability to understand and apply the basic concepts of group theory.

Transferable Skills:
Recognition of the power of mathematical abstraction and the importance of mathematical rigour. Critical thinking and problem-solving skills. Ability to present work in a scholarly manner.

1. Groups, abelian groups, subgroups, generators, homomorphisms, isomorphisms, kernels, images, conjugacy, cosets, normal subgroups and quotient groups.
2. Fundamental group theorems, including Lagrange’s theorem, the first isomorphism theorem and Cayley’s theorem.
3. Examples of groups, including cyclic, dihedral and symmetric groups.
4. Concepts related to the symmetric group, including cycles, transpositions and signs.

An interactive version of this list is available at https://whelf-cardiff.alma.exlibrisgroup.com/leganto/public/44WHELF_CAR/lists/7831546920002420?auth=SAML