Visual Algebra, Lecture 5.2: Five features of group actions

Описание: Every group action has five fundamental features that are worth trying to understand: orbits, stabilizers, fixators, the kernel, and the fixed points. The first three of these are “local”---they depend on individual set or group elements, and the last two are “global”---they depend on the homomorphism that defines the action. Several of these are “dual” to each other. For example, the stabilizer of a set element are the groups elements that fix it, whereas the fixator of a group element are the set element that it fixes. We will interpret all of these features in terms of action graph visual and our "group switchboard" analogy. We’ll also see a new visual in this lecture, called a fixed point table.

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

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CHAPTERS

0:00 Introduction
1:14 Group actions, action graphs, and G-sets
4:25 Global vs. local features
6:03 Two local features: orbits and stabilizers
9:38 The third local feature: fixators
16:15 Our binary square example and the three local features
18:34 Orbit tables, and the duality of orbits and stabilizers
20:40 The stabilizer subgroup
25:54 Elements in the same orbit have conjugate stabilizers
30:44 Binary hexagon example
33:36 Two global features: fixed points and the kernel
37:23 Global features, action graphs, and group switchboards
39:36 Fixed point vs. kernel duality in the orbit table