Module 1 – An Introduction to Group Theory
“What is especially striking and remarkable is that in fundamental physics, a beautiful or elegant theory is more likely to be right than a theory that is inelegant. A theory appears to be beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of Mathematics we already have. Symmetry exhibits the simplicity.”
Nobel Prize Physics 1969
Group Theory is perhaps the most fundamental of all topics in Mathematics. A camp based in France, the land of Evariste Galois, feels obliged to give young mathematicians, still in high school, access to these amazing concepts and ideas. An insight into proof, underlying structure and the type of Mathematics found in university courses.
Whilst conceptually challenging the actual Mathematics involved is relatively straight forward and so the prerequisite knowledge for this module is quite simple. All that is required is an openness to new ideas, enthusiasm, curiosity and the willingness to get lost in tough concepts.
In this module students will be introduced to Group Theory from the ground up. Symmetry is the basis of Group Theory and the module starts with a discussion of how we can describe symmetry, whether we can have degrees of symmetry, and how different symmetries relate to each other.
During the first few lessons students will also receive a brief introduction to Modular Arithmetic, Equivalence Relations, Complex Number Arithmetic, as well as a brief treatment of basic Matrix Arithmetic. The students will then be in a position to appreciate the definition of a Group.
With these tools students will go onto discover how the symmetry in geometry can be related to permutation. This will lead to the discovery of and other structural relations in mathematics that can be described in Group Theory. This will be formalized in the definition of isomorphism, subgroups and cosets.
The final destination of the module will include Lagrange’s Theorem and a deeper dive into structure with an introduction to Normal subgroups, Quotient groups and Homomorphisms.
It will be challenging. But if you like pure mathematics, this will be a lot of fun.
The module will include lectures, activities, tasks and exercises. The students will decide the pace that they need to go at so as to really appreciate the beauty of the subject. Be ready to ask a lot of questions.