We begin this lecture with two qualitative observations: (1) elements with larger orbits tend to have smaller stabilizers, and (2) actions whose fixed point tables have more checkmarks tend to have more orbits. We will then quantify each of these into a formal mathematical theorem. The orbit-stabilizer theorem gives a bijection between elements in an orbit, and cosets of the stabilizer. The orbit-counting theorem says that the average size of a fixator is the number of orbits. As before, our favorite visuals---action graphs, group switchboards, and fixed point tables, will be crucial in understanding what these theorems are really saying, and how to prove them.
Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule/visualalgebra/
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CHAPTERS
0:00 Introduction
1:20 Qualitative observation 1: larger orbits have smaller stabilizers
2:39 Qualitative observation 2: more checkmarks means more orbits
3:37 Two theorems on orbits
5:52 The orbit-stabilizer theorem
7:17 The bijection between cosets and elements in the orbit
10:28 The orbit-stabilizer theorem in our binary square example
12:54 Proof of the orbit-stabilizer theorem
20:37 The orbit-counting theorem & fixed point tables
24:41 Proof of the orbit-counting theorem