Wednesday, February 14, 2024


Negotiations are everywhere - at war, in trade, in personal relationship, even in relationships with yourself. Hereby I list some hints as they come to mind either internally or from external sources.

Negotiations process is a poker game. So same hints apply. Another science is mathematical game theory, however this one is way more sophisticated. 

Don't trust. War is an art of lies - Sun Tzu. 

Less talking, more questioning. Ask as many questions as you can, get as much information as you can.

Don't insult. Threat can also be an insult so use it only as a last remedy. If the other side isn't afraid to lose it can fight till the end just because it is insulted.

Less proposals, more inquiries. Let the other side make a proposal first so you know their limits and you have more information. Make your own proposal only in the end.

Plan B. Always have plan B for any step. Especially for such dangerous steps as proposals and threats. What if you proposal is rejected, what if they don't care about your threat.

Break down demands into small parts. If there are multiple demands, one of them is always major. So we need to find which one.

Don't be afraid. Turn off your fear. Fear prevents us from making good decisions. Use logic instead.

Don't take a decision under pressure. The other party might make a rush. We need to distinct serious ultimatum and alleged rush.

Saturday, February 10, 2024

Cruelty to animals

Vegans are concerned about cruelty to animals. They refuse to eat animal food because animals are killed in farms. Does it make sense? Yes and no.

Cruelty should be divided in 2 parts. First is cruelty for necessity. So it is somehow justified. The second is opposite - cruelty without necessity hence not justified.

Justified cruelty

Predators need to kill other animals to get food and survive. Wolf eats deer, fish eats other fish, insects eat other insects, even some plants are able to kill and eat small insects. Anyway, each one of us kill and eat bacteria and microorganisms even though we don't want and don't know about that.

Humans kill animals at farms to eat them. Even though I personally don't eat meat, this type of cruelty still can be considered as justified.

While refusing animals meat, vegans eat insects, seafood, plant roots and leafs. Plant doesn't want us to eat it's roots because it will die without roots. Roots is a crucial part of the plant body, it serve for digestion of water from the ground. 

Neither plants want us to eat their leafs. Leafs play important role in the breathing process of the plant. 

The only thing the plant wants us to eat is its fruit. Fruit serves for populating area and thanks to it seeds are spread on a long distance around the tree. So this is the only thing we should eat if we want to be truly vegans.

Non-justified cruelty

If cruelty is applied with no valid human reason it is not justified. Examples are circuses, zoo, sterilization. To understand the level of cruelty ask yourself: would you be ready to treat your child in the same way? What would you feel if someone treated you in the same way?

Just imagine your child is beaten by the circus trainer just to get skill to entertain public. Imagine your child to be hold in a dirty cage. This is even worse than a prison.

Imagine someone told you that you must be sterilized just to prevent overpopulation of similar creatures like you. What would you feel?

This is the way we treat animals. At the same time we lie that we love them. It is not love.

We bring our children to zoo and circuses to entertain them. What a fun. We teach cruelty since the childhood. And this cruelty is not justified. It will never be.

Thursday, February 8, 2024

Society of excessive consumption

We complain about poverty and ecology problems, however at the same time we contribute to them by over consumption.

Examples are:

Cutting grass in the yard. Is there any value of doing that apart from that yard "looks cute and clean"? Greens produce oxygen. If greens are cut, they lay down and become rotten producing methane which destroys our atmosphere.

Cosmetics and perfume manufacturing. Any girl spending twenty bucks for a gym looks much more cute than the girl spending the same money on cosmetics. Cosmetics manufacturing harms ecology.

Gym and sports. I don't understand people paying for gym and taking taxi or using elevators at the same time.

Cleaning supplies, powders etc. Is there any value of using them apart from that clothes look cute and clean? If you want to kill all the microbes on the planet with chemicals, you need to kill all the life on the planet. Chemicals leftovers after washing machine go to rivers and oceans. They poison fish and we eat this fish afterwards.

Home renovation. 

Spices, cooking food. These artificially stimulate our appetite which results in overeating, not good for health.

Rank and gender needs

Cosmetics, luxurious products, everything that make you look "beautiful" is the result of the need for higher status (rank). Males and females constantly fight for attention of each other - it is very natural.

To attract a partner of the opposite sex, one need to show its higher status (rank) in the pride. The higher your rank the more attractive you are for potential sex partners. We need to attract sex partners because it is our way to populate our genes over the planet, it is our instinct.

Big animals use power to demonstrate their status. Smaller animals like birds have bright colors of the body to attract opposite gender.

Humans didn't go far from that. Some of us use "fair" ways to demonstrate our attractiveness. We do sports to become more healthy, strong and hence attractive.

In the current society, not only physical gorilla power demonstrates our rank. Country president could be more attractive to women that a good athlete. The one's rank is determined by his or her position in the society, money power, carrier etc. That's why we study, fight for high paying jobs, make business.

However what the one could do if he or she has nothing to increase his or her rank in a "fair" way? What if you are not an athlete, you are not intelligent to get a good job?

Right. If the product is selling badly, invest in marketing and advertising. Doesn't matter what is inside the box, we will make the package look perfect. This is the easiest way to attract a potential buyer.

We will apply cosmetics, we will make photos with the background of luxury cars, we will dress on attractive dressings etc. All of these are redundant spendings for the society and these harm planet ecology. But these help to increase ranks of those species with lower statuses.

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?


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.


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? 


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.
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? 


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


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.


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?


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?


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?


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


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?


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.


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


Exercise 4.7


Exercise 4.8

Reference to the Exercise 2.4 and 1.1:

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.

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:

Reference to the Exercise 2.7 and 1.14 part (d)

Reference to the Exercise 2.8

Chapter 5. Five families


Answers to the book's exercises

Chapter 6. Subgroups


Answers to the book's exercises

Chapter 7. Products and quotients


Answers to the book's exercises

Chapter 8. The power of homomorphisms


Answers to the book's exercises

Chapter 9. Sylow theory


Answers to the book's exercises

Chapter 10. Galois theory


Answers to the book's exercises