Matt Macauley Profile picture
Mar 29, 2022 23 tweets 10 min read Read on X
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
Next, we asked when a group G is isomorphic to a direct product or semidirect product of its subgroups, N & H.

Here are two examples of groups that we are very familiar with.

3/14 M
Say that subgroups N,K are LATTICE CONJUGATES if
1. G = NK, and
2. N∩K={e}.

This has a very nice interpretation in terms of subgroup lattices! (Recall: NH is only a subgroup if one of N & H normalizers the other. Or weaker: one of them is normal.)

4/14 M
Suppose N,H are subgroups of G. We proved the following.

Thm 1: G≅N×H iff N & H are lattice conjugates, and BOTH normal.

Thm 2: G≅N⋊H iff N & H are lattice conjugates, and N is normal.

This is easy to check in the subgroup lattice by inspection!

5/14 M
Here's an example. The semidihedral group SD_8 breaks up into a non-trivial semidirect product of two of its subgroups, two different ways.

We can see this by inspection, both in its subgroup lattice (see the lattice conjugates?) and in its Cayley diagrams.

6/14 M
Here's a non-example. Recall the generalized quaternion group Q_{32} = <ζ_{16}, j>, where r=ζ_{16} = e^{2πi/16}. Note how every nontrivial cyclic subgroup contains r^8=-1.

Q_{32}, and more generally, Q_{2^n}, is NOT a (nontrivial) semidirect product of two subgroups.

7/14 M
I like this example a lot. The dihedral group D_6 is isomorphic to a direct product, D_3×C_2, and a semidirect product C_6⋊C_2. The Cayley diagrams are shown on the right. Do you see each pair of lattice conjugates in the subgroup lattice?

And that's all for Chapter 4!

8/14 M
Next up, Chapter 5: Group Actions. Let's go back to the first week, and recall the difference b/w a GROUP of ACTIONS, and the SET of CONFIGURATIONS it permutes.

9/14 M
Notice how there was a bijection b/w actions and configurations.

In this section, we'll drop this requirement. In doing so, we'll get "action diagrams", that are generalizations of Cayley diagrams.

But how do we just drop this requirement? Can you think of an example?

10/14 M
Here's one! My "favorite example" of a basic action, and it embodies many properties that we'll see throughout this chapter.

I struggled mightily with group actions as a student. MOST STUDENTS DO, THIS IS NORMAL.

The algebraic definition is just so weird & unmotivated.

11/14 M
Of course, if you'd asked me as a student if I understood actions, I'd have said "yes", "I think so", or "pretty much".

Looking back, I really didn't. I think EVERYONE who has studied math can in some way relate to this feeling.

Here's how I explain group actions now.

12/14 M
Formal definition: a group G acts on a set S if there is a homomorphism

φ:G-->Perm(S)

to the set of permutations of S.

φ(g) = "the permutation that results by pressing the g-button". That's it!

13/14 M
Compare that to the unmotivated, classic algebraic definition. Instructors are the "gatekeeper" of this mysterious concept, which isn't appreciated by most students.

Our fun w/ group actions is only getting started. Stay tuned for classic concepts revealed in new ways!

14/14 M
We started out Wednesday with three standard examples of group actions involving D_4. We've seen pictures like these earlier, but only now that we've defined group actions can we speak of them in the proper framework.

1/9 Wed 🧵👆👇
Every action has FIVE FUNDAMENTAL FEATURES.

Three "local" properties, of individual set or group elements. (today)

Two "global" properties, of the action (homomorphism) φ:G→Perm(S). (Friday)

We'll soon see a certain duality relating pairs of these.

2/9 W
The 3rd local feature isn't always emphasized as much in a class as orb(s) and stab(s) are, but it's important, and is in some sense "dual" to the latter. We'll see what I mean by "dual" via a "fixed point table".

Note: I write LOCAL features in lowercase.

3/9 W
I strongly recommend thinking about these features, first and foremost, using our "group switchboard analogy" from #Week10.

It makes the proofs straightforward, like that stab(s) is a subgroup. We talked it through in terms of switchboards, but the proof is left for HW.

4/9 W
Let's return to my favorite example.

Key observation: the BIGGER an orbit gets, the FEWER paths loop back. So the stabilizers are SMALLER.

Notice how |stab(s)|*|orb(s)|=|D_4|=8 in each case.

WHERE HAVE WE SEEN THIS BEFORE?!?!

5/9 W
The "ah-ha's" were quite audible at this moment. Remember this; I've been hammering it home for over a month:

The INDEX of N(H) is the SIZE of cl(H).
The INDEX of C(g) is the SIZE of cl(g).

Both of these are special cases of a more general result about group actions!

6/9 W
I had skipped the proofs of both of these (only did a sketch of one, left the other for HW) because the proof of the more general result (the orbit-stabilizer theorem, which we'll see on Friday), is completely analogous.

7/9 W
Here's a key concept, I call it a "fixed point table". I think it should be self-explanatory.

Notice how the stabilizers and fixed point sets are "dual" to each other: one can be read off the columns, and the other off the rows.

We're going to revisit this table!

8/9 W
Note that elements in the same orbit might have different stabilizers, but not "too different". In fact, any two stabilizers must be conjugate subgroups. This picture illustrates why. I LOVE the simple "loop" interpretation on the right.

On Friday: the 2 global features!

9/9 W

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More from @VisualAlgebra

May 28, 2022
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.

In #Week2 of #VisualAlgebra, we explored the Rubik's cube, more Cayley diagrams, group presentations, the impossibility of the word & halting problems, and we classified all frieze groups.

Read 23 tweets
May 14, 2022
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
For example, how many of the following are possible?

Before reading on, see if you can answer this, and generalize to arbitrary groups.

There's a simple elegant answer, that I was never aware of. And I suspect that the majority of people who teach algebra aren't either.

3/12
Read 13 tweets
Apr 27, 2022
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 ImageImage
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
Some, but not all of these hold in general integral domains. This is what we'd like understand!

If b=ac, we say "a divides b", or "b is a multiple of a".

If a | b and b | a, they're "associates", written a~b.

HW: a~b iff they differ by a unit (i.e., a=bu).

3/8 M
Read 14 tweets
Feb 23, 2022
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 ImageImage
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 ImageImage
Next, we proved that if [G:H]=2, then H is normal. Here's a "picture proof": one left (resp., right) coset is H, and the other is G-H.

3/8 M Image
Read 36 tweets
Jan 20, 2022
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
Image
Image
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 Image
I showed 3 different groups of order 8, and asked if any are isomorphic. At this point, all they know about what that means is that two groups must have identical Cayley diagrams *for some generating set*.

3/8 W ImageImageImage
Read 16 tweets
Jan 14, 2022
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
For further insight, consider the symmetries of a triangle. This motivates the idea of relations, and why this "group calculator" tool is useful.

I got so many really good questions and comments in class today. How often does that happen on *Day 1* of abstract algebra??

3/4
Read 4 tweets

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