Visual Algebra, Lecture 5.4: Examples of group actions скачать в хорошем качестве

Visual Algebra, Lecture 5.4: Examples of group actions

In this lecture, we will explore four important actions: groups acting on themselves by multiplication, on themselves by conjugation, on their subgroups by conjugation, and on their cosets by multiplication. For each of these, we will characterize the five fundamental features, and the consequence of our two orbit theorems. In some cases, this will even lead to new theorems, such as a one-line proof of Cayley's theorem, or formulas for the number of conjugacy classes of elements and subgroups. The content in this video is a great way to practice the theory that we've learned thus far in this chapter. Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule... ------------------------------------------------------------------------------------------------------------------------------------------------------ CHAPTERS 0:00 Introduction 1:12 Groups acting on elements, subgroups, and cosets 2:47 Groups acting on themselves by multiplication 4:29 A 1-line proof of Cayley's theorem 7:05 Groups acting on themselves by conjugation: 5 features 9:26 Groups acting on themselves by conjugation: 2 theorems 10:33 Action graph of D₆ acting on itself by conjugation 12:28 Fixed point table of D₆ acting on itself by conjugation 13:15 Cosets of cyclic subgroups of D₆ and centralizers 16:02 Groups acting on their subgroups by conjugation: 5 features 18:19 Groups acting on their subgroups by conjugation: 2 theorems 20:23 Action graph of D₃ acting on its subgroups by conjugation 22:31 Cosets of D₃ subgroups and normalizers 25:05 Action graph of A₄ acting on its subgroups by conjugation 25:56 Fixed point table of A₄ acting on its subgroups by conjugation 26:38 Groups acting on themselves by multiplication: 5 features 31:30 Collapsing left cosets vs. right cosets in a Cayley graph 35:03 Every transitive action is equivalent to an action on cosets 39:28 A table summarizing our four actions & their features