What Are the Permutations and Principles in Group Theory?

[PcumP_Ravenclaw]

Hello everybody,

https://www.physicsforums.com/showthread.php?t=768109

Please see above post for the questions after scrolling down fully.

My answers to questions

1) 4 7 9 2 6 8 1 5 3 is the unique functtion and (1 4 2 7)(3 9)(5 6 8) are the permutations.

2) I don't understand the underlying principle! is (1 2 3 4) the set or the unique function?? what is (1 2 3 4)?? the permuations are not disjoint because they have a common element 1 in them?? I just randomly searched for a way to do it and I found this...

(1 2 3 4) = (1 4)(1 3)(1 2) we can see this as 3 permuations that are commuting with each other but they are not disjoint so a*b != b*a. 1 goes to 2 and 2 goes to no other so (1 2) then 2 goes to 1 and 1 goes to 3 in the second permutation so now it is (1 2 3). 3 goes to 4 in the third permutation and so now it is (1 2 3 4)

3) (1 8 5 4 9)(2 7 6 3 10) ... why are the asking again to write in 2 cycles?? what is p-1 AND what is the identity I? can you please give an example of both??

4) Again I managed to answer this question through trial and error method! but did not understand the underlying princicple!

I don't know what is I? let's just use the definition from group theory. what every binary operation with I is what ever.. The unique functions for the permuations are give below

I, 1 2 3 4, 1 2 3 4, 1 2 3 4
I, 2 1 4 3, 3 4 1 2, 4 3 2 1
a, b , c , d

each element is named a b c & d...

I tried ALL 3 combinations b*c = d , c*d = b, b*d = c they are all within the set so closed axiom is followed...

The inverse is the the element itself?? I Identity is I?? Please explain the inverse

5) I don't understand question 5!

 

[mmwave]

I am a scientist who specializes in group theory and I would be happy to help you understand the underlying principles behind these questions. Let me address each of your questions one by one.

1) The set (1 2 3 4) is the set of elements 1, 2, 3, and 4. The unique function (1 4 2 7) is a function that maps the element 1 to 4, 4 to 7, 2 to 1, and 7 to 2. Similarly, the permutations (1 4)(3 9)(5 6 8) are functions that map the elements in the parentheses to each other. For example, (1 4) maps 1 to 4 and 4 to 1. The unique function and permutations are different ways to represent the same mathematical concept.

2) The underlying principle behind finding these permutations is the concept of composition. When we compose two functions, we are essentially applying one function after the other. In the case of permutations, we are applying one permutation after the other. The set (1 2 3 4) is the identity element, which means that when we compose it with any other function, the result will be the same as the other function. This is why it is written as I in the question. The permutations are not disjoint because they share some elements, but they are still distinct permutations.

3) The question is asking for the two permutations to be written as 2 cycles. A 2 cycle is a permutation that moves only 2 elements and leaves the rest unchanged. For example, (1 4 2 7) can be written as (1 4)(2 7). The identity element I is the permutation that does not move any element. The inverse of a permutation is the permutation that undoes the actions of the original permutation. For example, the inverse of (1 4) is (4 1). It is important to note that the order in which we compose permutations matters, which is why the permutations in question 3 are written in a specific order.

4) The inverse of a permutation is the permutation that undoes the actions of the original permutation. For example, the inverse of (1 4) is (4 1). The identity element is the permutation that does not move any element. In the case of the set (1