A group action occurs when a group naturally permutates a set of states or configurations. Group actions are one of the most important topics in all of group theory, but it’s very often just not well understood by students taking algebra. I’ve read a ton of posts by people who just don’t quite get it, and are asking for help. You can easily find these posts too. And here’s the thing: if that’s you, you’ve come to the right place. What I’m going to show you in this chapter, really is, in my obviously biased opinion, the most intuitive and natural way to think about groups actions. Like I do with groups, when it comes to actions, I do things a little differently, but I’ll leave you convinced that they’re absolutely right. For example, I emphasize the concept of a G-set, which is often completely skipped, but that’s a huge mistake. We’ll talk about “group switchboards”, “action graphs,” and “fixed point tables.” Many of the visuals and analogies that you’ll see her are unique to my book and video series. Hopefully, not for long, though. I really think that everyone is a going to get a lot out of this chapter, from the novice student, to the Ivy League PhD candidate, to the seasoned instructor who is trying to better communicate these ideas to a class with diverse mathematical backgrounds. Also, let me warn you now: once you start thinking about things in this new way, you can’t go back! But that’s a good thing. I’m excited to show you how I think about groups actions, and I think you’re really going to like it too.
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
1:48 Overview
3:40 Actions vs. configurations
5:08 Action graphs and Cayley graphs
7:06 Our running example of 7 binary squares
11:08 The group switchboard analogy
14:33 G-sets: sets endowed with an algebra structure
20:30 Group switchboards and permutations
21:16 Group actions and the symmetric group
22:39 Transitive G-sets and cosets
28:28 Three actions of D₄: by multiplication and conjugation
31:10 Left vs. right actions
36:22 An alternative definition of a group action