This week, I recorded 4 new #VisualAlgebra YouTube videos on group actions. I heavily use the following concepts:

1. G-sets
2. Action graphs
3. Group switchboards
4. Fixed point tables.

Interested? I'll give you a preview here. Please share!

🧵👇1/16 If G acts on S, via ϕ:G→Perm(S), imagine that G has a "group switchboard", w/ a button for each element. Pressing it permutes elements of S, with the rule:

"Pressing the a-button followed by the b-button is the same as pressing the ab-button." This means:

ϕ(a)ϕ(b)=ϕ(ab)

2/16

Now that my #VisualAlgebra class is in the books, I want to post a long "meta thread" of all 16 weekly threads, with daily summaries. Here's my entire class, including lectures, HW, & exams, in one convenient place.

And stayed tuned for some surprise announcements below!👇🧵 We started #Week1 of #VisualAlgebra with a few quotes from "A Mathematician's Apology" on the beauty of mathematics, and then saw Cayley diagrams for the symmetries of the rectangle and triangle.

https://twitter.com/VisualAlgebra/status/1482065699212120064

I woke up a few days ago with the sobering realization: actually, I do NOT really understand groups actions.

Spoiler: I do now, but it took some work. And now I realize how incomplete my understanding was. 😳

Let me explain, I think some of you might enjoy this!

1/12 🧵👇 See those "orbit diagrams" above? I got to thinking: "how can we characterize all possible diagrams?" Equivalently, all transitive actions of D_4 (or a group G in general).

Playing around with things, I came up with a few more. But I still didn't know the answer. Do you?

2/12

Finishing up our🧵👇 #VisualAlgebra class in #Week15 with divisibility and factorization. I'm a little short on visuals, but here are two really nice ones on what we'll be covering, made by @linguanumerate.

Henceforth, we'll assume that R is an integral domain.

1/8 Mon The integers have nice properties that we usually take for granted:

--multiplication commutes
--there aren't zero divisors
--every nonzero number can be factored into primes
--any 2 nonzero numbers have a unique gcd and lcm
--the Euclidean algorithm can compute these

2/8 M

We started #Week11 of #VisualAlgebra with a new diagram of one of the isomorphism theorems. I made this over spring break. The concept is due to Douglas Hofstadter (author of "Gödel, Escher, Bach"), who calls this a "pizza diagram".

1/14 Mon 🧵👇 Though we constructed semidirect products visually last week, we haven't yet seen the algebraic definition. On Friday, we saw inner automorphisms, which was the last step we needed.

Recall the analogy for A⋊B:
A = automorphism, B = "balloon".

2/14 M

We started #Week7 in #VisualAlgebra yesterday with the tower law.

Here are two ways to think about it. One involves cosets as "boxes" in a grid, and the other is in terms of the subgroup lattice: to find the index [H:K], just take the product of the edges b/w them.

1/8 Mon Pause for a quick comment about cosets in additive groups. Don't forget to write a+H, rather than aH. Here's a nice way to see the equality of a left coset and a right coset.

2/8 M

WEEK 2 of #VisualAlgebra! This is only Lecture #2 of the class.

Monday was MLK Day, but on Wed, we learned about the Rubik's cube! I got to show up my rare signed cube with Ernő Rubik himself from 2010! Did a shout out to @cubes_art's amazing talents.

1/8 W

We learned some neat facts about the Rubik's cube, like how the group has just 6 generators, but 4.3 x 10^{19} elements, and a Cayley diagram with diameter of 20 or 26, depending on whether you count a 180 degree twists as 1 or 2 moves.

2/8 W

WEEK 1: first lecture of #VisualAlgebra.

"The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way."
― G.H. Hardy, A Mathematician's Apology

1/4 We have not yet defined a group. Rather, we are exploring the intuition of them via symmetries. This will *motivate the axioms*, rather than the other way around.

What properties does this group have what might not hold more generally?

2/4

Happy New Year #MathTwitter! Let's start 2022 w/ Part 1 of a fun series: "Groups you Never Knew Existed...and others you can't POSSIBLY live without!"

Today we'll see the "diquaternions", a term you've never heard of b/c I made it up last month. Let's dig in! 🧵👇

1/17 We'll start with the familiar quaternion group Q_8. Shown here are several Cayley diagrams, a Cayley table, cycle diagram, subgroup lattice, its partition by conjugacy classes, and an action diagram of Aut(Q_8). Each of these highlights different structural features.

2/17

What does it really mean for a group to be "nilpotent"?

This year, I've asked many people to describe it in simple, memorable terms, and have yet to get a good answer.

Usually: something something about an ascending series. But what exactly, and WHY? Let's dig in! 🧵👇

1/17 First, I'm wasn't at all picking on anyone, but rather, at how this concept (and so many others) are taught in nearly every algebra class and book.

By the end of this thread, you'll learn what nilpotent really means in a memorable visual way you'll never forget!

2/17

Last week I did the Sylow theorems in class, and I want to share how I do them with my visual approach to groups.

To start, here are the 5 groups of order 12. Note how there are "towers of p-groups", for p=2 and p=3.

This is what the 1st Sylow theorem guarantees.

1/17 The key lemma needed for the Sylow theorems is:

"If a p-group G acts on S, then |Fix|≡|S| mod p."

Here's a picture proof of that, adapted from @nathancarter5's fantastic "Visual Group Theory" book.

2/17