Visual Algebra, Lecture 5.8: Simply transitive actions

Описание: A group action, or G-set it defines, is transitive if its action graph is connected. An action is free if the stabilizer of every element is trivial. This means that it is as “uncollapsed as possible.” Actions or G-sets that are transitive and free are called simply transitive. In this lecture, we’ll see that every simply transitive action is isomorphic to a group acting on itself by multiplication. In other words, the simply transitive action graphs are precisely the Cayley graphs. Then, we’ll see a number of examples of simply transitive actions, mostly from tilings of polygons, polytopes, the xy-plane, and even the hyperbolic plane.

Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule...

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0:00 Introduction
1:03 Transitive, free, and faithful actions
3:32 An action that is free but not transitive
4:55 Examples of simply transitive actions
6:24 Every simply transitive action graph is a Cayley graph
12:23 A simply transitive action of ℤ×ℤ
14:00 Simply transitive actions of D₃ and D₄
18:18 From finite to affine reflection groups
19:03 The affine Weyl group of type Ã₂
21:44 The affine Weyl group of type C̃₂
25:10 Weyl groups and Dynkin diagrams
28:31 The affine Weyl group of type G̃₂
29:54 Coxeter groups and affine Weyl groups of higher rank
33:52 Hyperbolic Coxeter groups
36:15 A simply transitive action of PSL₂(ℤ)