Tuesday, February 6, 2024

To the book "Visual group theory" by N. Carter: abstract algebra exercises, questions, ideas...

These thoughts came to me when I studied abstract algebra. Some of them are general and some are related to the certain book. I will update this post as I continue studying: so currently it could look crappy and uncompleted. You are welcome to discuss these thoughts in the comments section below. My current level of abstract algebra knowledge is pretty low, so I would be glad to any corrections or suggestions..

"Visual group theory" is the book I used to study abstract algebra from scratch. Below in the "Questions" section I include thoughts that came to my mind and not listed in the book. I didn't get answers and solutions to all my questions so far. Also I list some "Answers to the book exercises" in the consequent sections that are not given by the author.

Chapter 1. What is a group?

Questions

Can we apply any generator to any achievable Rubik's cube state?

By "any achievable state" we mean any cube condition that can be derived from the starting point (with same colored edges) via any combination of generators.

Solution:

We should be able to apply any generator to any state. This is because in Definition 1.9 Rule 1.8 dictates "Rule 1.8. Any sequence of consecutive actions is also an action."
The plain text of the rule seems like ambiguous. However it is verified by exercise 1.3, the answer to it is also given in the book.

Out of rules 1.6-1.8, is there any one redundant?

I. e. can some rule be derived from others? Are all of them necessary?

If generators are reversible, are combined actions reversible?

Generators are simplest non-breakable actions. Combined actions are any actions created by consequently applying any consequence of generators.
Let's assume Rule 1.7 means "every generator is reversible" instead of "every action is reversible". Does it mean that every action is reversible because every generator is reversible? 

Solution

Yes it does. Let's take some random action G bringing us from the node A⁰ to Aⁿ. This action is the result of applying n generators as follows:
generator g⁰ brings us from A⁰ to A¹,
generator g¹ brings us from A¹ to A²,
...
generator gⁿ brings us from Aⁿ-¹ to Aⁿ.
Let's denote reverse elements for generators as h.
Then 
h¹ is inverse for g¹,
h² is inverse for g²,
...
hⁿ is inverse for gⁿ.
Since every generator is reversible, we can apply h elements. Let's apply hⁿ to Aⁿ,...,h¹ to A¹. We can prove via mathematical induction that this gives us A⁰ in the end. Coming from Aⁿ to A⁰ is an inverse action to G.

Does a group comply with commutative property?

Does the order of actions performance affect the result? 

Solution:

In general, groups do not comply with commutative property. E. g. see the Exercise 1.4(b).  

Answers to the book's exercises

Exercise 1.13

It is about whole numbers, however the exercise says: "...we might name them things like "add 1" and "add -17...". However "-17" seems like not a whole number. So if we assume that the exercise means whole numbers, than it is not a group. We cannot satisfy the requirement of reversibility (Rule 1.6).

If the exercise is about integers than it is a group and we are in compliance with all the rules 1.5-1.8. Than the smallest set of generators is {add -1; add 1}.

Chapter 2. What do groups look like

Questions

How to present Cayley diagram as a table?

Graphical Cayley diagram is good for visualization. However a table is a good tool of exploration and automation of calculations.

Solution:

Rows names set (x-axis) will be a set of object states which is a set of all the nodes in Cayley diagram. Columns names set (y-axis) is generators (simplest non-breakable actions).

Can we get from any point to any point in Cayley diagram of a group?

We are given a group and its Cayley diagram. Let's choose its random node (point). Can we make a path via generators actions to any other node? Can there be a node belonging to a group that is not accessible from our chosen node?

Solution:

It is assumed that we can get to any target node from so called starting node by applying generators actions. Otherwise we wouldn't include that target node in a group.
E. g. in the Chapter 1 starting node was Rubik's cube with each edge having same color.
According to Definition 1.9, every action is reversible. Hence, we can get from any node to the starting node. In turn supra we noted that we can get from the starting node to any node.

If we could get from any node to any other node does it mean that every action is reversible?

Say we have a group defined as follows:
Rule 1.5. There is a predefined list of actions that never changes.
Rule extra. We can get from any state (node) to any other node.
Rule 1.7. Every action is deterministic.
Rule 1.8. Any sequence of consecutive actions is also an action.
This is the Definition 1.9 with one rule replaced. We removed Rule 1.6 "Every action is reversible" and we placed Rule extra instead "We can get from any state (node) to any other node". By "node" we understand here any state which is the result of any combination of generators applied to starting point.
Does it mean that Rule 1.6 should follow from these?

Solution:

Yes it follows. Let's take 2 random nodes: A and B. Rule extra means that we can always get from A to B and from B to A. Hence, A<->B action is reversible.

What if we choose another node for start?

Let's change the initial element, so we start from any other node of the Cayley diagram. The diagram per se remains the same.
Will it still be a group? Will we still comply with all the group rules? 

Answers to the book's exercises

Exercise 2.6

Let's deem Exercise 1.13 is about integers not whole numbers. Than Cayley diagram can be presented as follows. We draw an infinite straight (or not straight) line with all the integers marked on it. Each integer will have infinite outbound and infinite inbound arrows. Every integer will be connected with every other integer by 2 arrows directed in opposite ways.

Chapter 3. Why study groups?

Questions

What is the real practical use of groups?

In chapter 1, we saw that combinations of Rubik's cube can form something called a group. In chapter 3, we saw that the same can draw molecules crystal and pattern on a fence. But why do we need it? Can groups help us to solve Rubik's cube puzzle? Can they help us to see some crystal structure that we cannot see in the microscope

Answers to the book's exercises

Chapter 4. Algebra at last

Questions

Can number of columns in a multiplication table be lower?

Multiplication tables are described in the Section 4.3 of the Carter book. It is defined in the book, that number of rows in a table is equal to number of columns which is the total number of actions. However to define a table in a unique way we need only generators as columns. So we don't need to include all the actions in columns.

How can creation of multiplication tables be automated?

It is obvious that we can write a customized piece of code for every certain group. But is there a universal algorithm that can be applied to any group? So we don't customize it every time?
If we talk about certain group - can we automate it via formulas in Google Sheets or MS Excel? This is much easier than writing code.

Are all 4 of the Definition 4.2 clauses necessary?
Out of clauses 1-4, is there any one redundant? I. e. can some property be derived from others?

Can we take any node as a starting point in the diagram of actions?

When creating a diagram of actions, we need to choose the initial element. As per the Definition 4.1, we can choose any element. Why any? Will it still be a group as per Definition 1.9 regardless of which element we choose as a starting node?

Can 2 row headers of a multiplication table be the same?

Can the same node appear 2 times in one axis of a multiplication table?

Solution:

The answer is nope. Proof? Rule 1.7 provides that every action is deterministic. Let's prove that any consequence of actions is deterministic either. To be continued...

For the same given group with the same starting node, can there be 2 different multiplication tables?

E. g. axis nodes (apart from the starting node) could differ, table contents could differ.

Solution:

Short answer is yes. Order of nodes in the axis could differ. If we denote a node by actions path, sometimes we could come to the same node via 2 different paths. See e. g. the solution to the Exercise 4.8 infra, references to the Exercises 2.5 and 1.4(b). We can come to the 321 node via 2 different ways: lrl and rlr.

Answers to the book's exercises

Exercise 4.6
(c)

abcdexyza2b2c2d2
aabcdexyza2b2c2d2
bbadcxezyc2d2a2b2
ccdabzyxed2c2b2a2
ddcbayzexb2a2d2c2
eeyxza2b2d2c2acdb
xxzeyc2d2b2a2bdca
yyezxb2a2c2d2dbac
zzxyed2c2a2b2cabd
a2a2d2b2c2acbdexzy
b2b2c2a2d2dbcayzxe
c2c2b2d2a2bdacxeyz
d2d2a2c2b2cadbzyex
nrrbrbrbrrrrrrbrbrbrrrrbbrbbr

Exercise 4.7

......-3-2-10123...
..............................
-3...-6-5-4-3-2-10...
-2...-5-4-3-2-101...
-1...-4-3-2-1012...
0...-3-2-10123...
1...-2-101234...
2...-1012345...
3...0123456...
..............................

Exercise 4.8

Reference to the Exercise 2.4 and 1.1:
pnnp
pnpnnp
npnppn

Reference to the Exercises 2.5 and 1.4(b):
Let's take the following notation:
We have 3 pictures denoted with the numbers 1, 2, 3. 
We denote states (nodes) as:
- the "123" state is: "1" picture is on the left, "2" picture is in the center, "3" picture is on the right;
- the "132" state is: "1" picture is on the left, "3" picture is in the center, "2" picture is on the right. 
- etc.
We denote generators actions as: 
- "n" is no action, an identity;
- "l" is swapping a picture on the left with a picture in the center;
- "r" is swapping a picture on the right with a picture in the center.
As it is more convenient, I draw multiplication table in 2 steps. First, I create a table with nodes names like "123". After that, I convert these names to actions like "lrl". Useful function to do that in Google Sheets is vlookup.
123132213231312321
nrllrrllrl
123n123132213231312321
132r132123312321213231
213l213231123132321312
231lr231213321312123132
312rl312321132123231213
321lrl321312231213132123
123132213231312321
nrllrrllrl
123nnrllrrllrl
132rrnrllrlllr
213lllrnrlrlrl
231lrlrllrlrlnr
312rlrllrlrnlrl
321lrllrlrllrlrn

Reference to the Exercises 2.6 and 1.13
If the Exercise 1.13 is about whole numbers than it has no solution, see also the answer to the Exercise 1.13 supra. If it is about integers than the multiplication matrix is as follows:
......-5-4-3-2-1012345...
..........................................
-5...-10-9-8-7-6-5-4-3-2-10...
-4...-9-8-7-6-5-4-3-2-101...
-3...-8-7-6-5-4-3-2-1012...
-2...-7-6-5-4-3-2-10123...
-1...-6-5-4-3-2-101234...
0...-5-4-3-2-1012345...
1...-4-3-2-10123456...
2...-3-2-101234567...
3...-2-1012345678...
4...-10123456789...
5...012345678910...
..........................................

Reference to the Exercise 2.7 and 1.14 part (d)
1-1
11-1
-1-11

Reference to the Exercise 2.8



Chapter 5. Five families

Questions

Answers to the book's exercises

Chapter 6. Subgroups

Questions

Answers to the book's exercises

Chapter 7. Products and quotients

Questions

Answers to the book's exercises

Chapter 8. The power of homomorphisms

Questions

Answers to the book's exercises

Chapter 9. Sylow theory

Questions

Answers to the book's exercises

Chapter 10. Galois theory

Questions

Answers to the book's exercises

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