Visual Algebra, Lecture 5.4: Examples of group actions

Описание: 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...

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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