5:29 a good analogy for this is stacking stones. The permutations of the set that are inside the brackets are like the action of taking away from or putting a stone onto the pile you're building, while the numbers on the number line represent the total number of stones in the pile. Another interesting thing about thinking of it this way is that when you add or subtract (permute) a set of numbers, the result that you get on the number line can be described as an infinite set of permutations to get each unique number or "position" on the number line, which we describe using the concept of place value so we only have to use 9 symbols.
This is important: The whole point of a group is that if you're given a set S containing elements x and y (among others) and an operator o, then x o y := z, where z belongs to S. This further implies that z may be x or y. At the end of the day, it's all a question of the definitions you create for each of the x o y compositions of your set. This matters because the set of linear permutations is an EXAMPLE of a group structure defined by the definitions you create. But there are other ways to create groups without using permutations. Here's a simple example. Given a set S = {A, B}, and an operator o, define: A o A := A A o B := A B o A := B B o B := B Then for any x,y E S, (x o y) o y = x o (y o z), and so {S, o} forms a group. This group looks nothing like a set of permutations, but it is a group. There are very few terms involved, so you can evaluate everything explicitly to convince yourself. In total, there are 8 ways to define associative operators for a set of 2 elements. This balloons to 113 ways for a set of 3 elements, and so on and so on (write a computer program to convince yourself of this). So, the point at the end of this tiny excerpt is that viewing groups as set of permutations of another set endowed with a linear mapping operator is pretty limited. Useful, for sure, but really limited. The permutation group is only 1 way of 8. So a group is not a set of permutation operators endowed with a linear mapping operator. Rather, a set of permutation operators endowed with a linear mapping operator is an EXAMPLE of a group. I hope this helps. -Float. _______ I have a challenge for you. Given a set S := {x_1, x_2, ... , x_w, ... , x_n} of n elements, we can define P(x_w) := w. That is, for some x_w in S, P(x_w) tells you the location of x_w in the ordered set S. Given some {x, y, Q} E S, Q := x o y, it follows (not proven here) that: P(Qxy) = Floor[k * n^(n*P(x) + P(y) + 1 - n^2)] mod n, such that 0
It will be fun to watch this dude proving Cayley's Theorem for groups, after he basically DEFINED them as sets of permutations of another set, which in this cause will be a tautology :)
@@bonbonpony He defined it as a set with composition and additional rules on top. If I were to guess how he's going to prove it, he's going to say something like "Remember how I told you to think of elements of a group as permutations of a set ? Well now we're going to prove that from the axioms."
The integers wouldn't form a group. The permutation of positive real numbers under multiplication would look like fixing zero in its place and stretching or squishing the real line by a factor while keeping everything evenly spaced. Just like in the case of addition we follow where 0 lands after the sliding, in multiplication we follow where 1 goes. If we include the negative reals as well then multiplication by a negative number would flip the orientation of the continuum (real line) besides stretching or squishing.
5:29 a good analogy for this is stacking stones. The permutations of the set that are inside the brackets are like the action of taking away from or putting a stone onto the pile you're building, while the numbers on the number line represent the total number of stones in the pile. Another interesting thing about thinking of it this way is that when you add or subtract (permute) a set of numbers, the result that you get on the number line can be described as an infinite set of permutations to get each unique number or "position" on the number line, which we describe using the concept of place value so we only have to use 9 symbols.
This is important:
The whole point of a group is that if you're given a set S containing elements x and y (among others) and an operator o, then x o y := z, where z belongs to S. This further implies that z may be x or y. At the end of the day, it's all a question of the definitions you create for each of the x o y compositions of your set. This matters because the set of linear permutations is an EXAMPLE of a group structure defined by the definitions you create. But there are other ways to create groups without using permutations.
Here's a simple example.
Given a set S = {A, B}, and an operator o, define:
A o A := A
A o B := A
B o A := B
B o B := B
Then for any x,y E S, (x o y) o y = x o (y o z), and so {S, o} forms a group. This group looks nothing like a set of permutations, but it is a group. There are very few terms involved, so you can evaluate everything explicitly to convince yourself. In total, there are 8 ways to define associative operators for a set of 2 elements. This balloons to 113 ways for a set of 3 elements, and so on and so on (write a computer program to convince yourself of this).
So, the point at the end of this tiny excerpt is that viewing groups as set of permutations of another set endowed with a linear mapping operator is pretty limited. Useful, for sure, but really limited. The permutation group is only 1 way of 8. So a group is not a set of permutation operators endowed with a linear mapping operator. Rather, a set of permutation operators endowed with a linear mapping operator is an EXAMPLE of a group.
I hope this helps.
-Float.
_______
I have a challenge for you.
Given a set S := {x_1, x_2, ... , x_w, ... , x_n} of n elements, we can define P(x_w) := w.
That is, for some x_w in S, P(x_w) tells you the location of x_w in the ordered set S.
Given some {x, y, Q} E S, Q := x o y, it follows (not proven here) that:
P(Qxy) = Floor[k * n^(n*P(x) + P(y) + 1 - n^2)] mod n, such that 0
A group supposed to have left and right identity elements. But your example has two right identities and no left.
yOU ARE ONE GREAT TEACHER!
Keep doing videos for us for other chapters like real analysis as well as sequence....
This is one of the best explanations! Good job Nice work!
You're so cool ❤❤❤... A teacher whom everyone wants ✨
Thank you sir
i am glad that i found you
could you tell me the name of the theorem which states that we can look at all group elements as set permutations???
It's called Cayley's Theorem in case you're still looking for it after two years.
It will be fun to watch this dude proving Cayley's Theorem for groups, after he basically DEFINED them as sets of permutations of another set, which in this cause will be a tautology :)
@@bonbonpony He defined it as a set with composition and additional rules on top. If I were to guess how he's going to prove it, he's going to say something like "Remember how I told you to think of elements of a group as permutations of a set ? Well now we're going to prove that from the axioms."
Okay, right.
Cenk Uyger liked your comment.
So is the idea that each group element represents an index of one to one functions?
Love it
Hi, group theory is long eh?
It's definitely much longer when you interject "OK?" every couple of words :J
thanks
What would the permutations of integers look like under multiplication
The integers do not form a group under multiplication.
The integers wouldn't form a group. The permutation of positive real numbers under multiplication would look like fixing zero in its place and stretching or squishing the real line by a factor while keeping everything evenly spaced. Just like in the case of addition we follow where 0 lands after the sliding, in multiplication we follow where 1 goes. If we include the negative reals as well then multiplication by a negative number would flip the orientation of the continuum (real line) besides stretching or squishing.
integers won't form a group under multiplication because identity is defined but no number other than 1 and -1 has a inverse
thanks..
10:05, joke?
e ○ g ○ g = egg
Ö
okay?
Right.