Abstract Algebra - Fall 2022
Part 1 : Mini-course on group theory and the Rubik’s cube
Dates: October 2022
Faculty:
Diletta Martinelli
(University of Amsterdam)
Giovanni Placini
(University of Florence)
Daan Van de
Weem
(Tutor)
Dilletta Martinelli
Giovanni Placini
Daan van de Weem
Syllabus:
Textbooks:
Dummit and Foote, Abstract Algebra. Hernstein, Abstract Algebra
Janet Chen, Group theory and the Rubik’s cube.
Course Materials: coming soon.
Course Schedule :
Week 1 - Monday October 17:
Afternoon session: Introduction meeting (Giovanni +Diletta). Recap on the definition of a group, main examples. Basic properties. Abelian groups: examples and non-examples. Order of an element in a group.
Break
Distribute the Rubik’s cube. Familiarise with the cube. Cube notation. Upper bound for the possible configuration of the cube. Making the Rubik’s cube into a group.
Reference: Janet’s Chen up to page 13 (We stop before the notion of subgroup).
Week 1 - Tuesday October 18 (Diletta):
Morning session: Homomorphisms and Isomorphisms. Group actions. Details on Symmetric group.
Break
Afternoon session: Exercises, examples. Introduction on the configurations of the cube (sigma, tau, x, y). Group acting on the configuration set.
Reference: Section 1.6, 1.7 Dummit and Foote. + Example on symmetric group.
Week 1 - Wednesday October 19 (Giovanni):
Morning session: Subgroups. Examples. Cyclic groups
Break
Afternoon session: Exercises and Examples.
Reference: Dummit and Foote, Section 2.1, 2.2. What’s needed of Section 2.3.
Week 1 - Thursday October 20 (Diletta):
Morning session: Cyclic group. Subgroups generated by subset of a group. Exercises.
Break
Afternoon session: 3:30-4:30. Presentation of group projects. Advice on how to give a good presentation.
Reference: Dummit and Foote, Section 2.3, 2.4.
Week 1 - Friday October 20 (Giovanni):
Morning session: Quotient Groups, Coset, Lagrange Theorem.
Break
Afternoon session: Exercises
Reference: Dummit and Foote, Section 3.1, 3.2.
Week 2 - Monday October 24 (Diletta):
Morning session: Isomorphisms theorem. Direct product and semi-direct products.
Break
Afternoon session: Exercises and Examples.
Reference: Dummit and Foote, Section 3.3., 5.1, 5.5. What’s needed of Chapter 5.
Week 2 - Tuesday October 25 (Diletta+Giovanni+Daan):
Morning session: Q&A on the course material. Work on group assignments.
Break
Afternoon session: Q&A on group assignments.
Week 2 - Wednesday October 26 (Giovanni):
Morning session: Cubology laws. Valid configurations. Solvability of the cube. What’s left on the cube.
Break
Afternoon session: (Giovanni+Diletta+Daan) Q&A. Group assignments.
Reference: Janet’s Chen
Groups:
Group 1: Ayodeji, Hamza, Ammara, (Aneeqa). Cyclic groups. Tutor: Diletta
Group 2: Humaira, Tehmina, Mahsood, (Dildar). Class equation. Tutor: Giovanni
Group 3: Ali, Faraz, Awais, Umer. Cayley’s theorem. Tutor: Daan
Week 2 - Thursday October 27 :
Morning session: Presentation group 1 + 2. Feedback informal from each tutors.
Break
Afternoon session: Presentation group 3. Research outlook. Diletta, Daan, Sohaib, Giovanni.
Dinner with everyone.
Week 2 - Friday October 28:
Morning session: More Research outlook. Diletta, Daan, Sohaib, Giovanni. Career advice.
Break
Afternoon session: Colloquium talk by Diletta on the work of Maryam Mirzakhani. Then documentary about her life “Secrets of the surface”. Conclusion.
