Question

Please, explain me the table from the right hand side.
How to define the elements?
Why is the inverse the same with the original form?
What is the order?
http://www.youtube.com/watch?v=b2Ozprvt8cw&list=PLF379B0552AD17780

Answer 1

Each number represents a vertex (angle). Every operation is just a swapping of the vertices.

Start with:
1, 2, 3
If you swap 1 and 3, which can be written as (13) or as (31) you end up with:
3, 2, 1

If you preform the same swap a second time, you end up with:
1, 2 ,3
Any operation which gets you back to where you started for a previous operation could be considered an inverse operation.

Consider the lower operations:
1, 2, 3
When you apply (123) you get:
2, 3, 1
If you applied (123) again you would get:
3, 1, 2

This isn't back where we started, so (123) isn't the inverse of (123). The actual inverse in this case is (132).

The order of an operation is just the number of the operation is just the number of elements that get moved around in the shuffle. I'm not sure why "do nothing" has order 1 and not order 0 though.

Answer 2

The way to read an operation is as follows:
Consider (352)
We start with:
1,2,3,4,5

Next look at 35. This means put 5 where 3 was.
1,2,5,4,_
Next we look at 52. This means put 2 where 5 was.
1,_,5,4,2
Finally, at the very end we put our first number 3 where the remaining empty spot is.
1,3,5,4,2

Suppose we wanted to find the inverse. We would want to put 2 where 3 is, 5 where 2 is, and 3 where 5 is. This would make our operation (325).

Answer 3

1,2,3 apply (123) means 1--> 2 , 2 --> 3 and 3 --> 1
in other word, if i have a,b,c apply (123) I will have b,c,a. right?

Answer 4

That looks right.

Answer 5

I got it, thanks a lot. I am new in this field. Need a lot of help. :)

Answer 6

and how about the order? what does it mean?

Answer 7

Order is number of swaps preformed.

Answer 8

I mean, number of elements moved, rather.

Answer 9

if I have 4 nodes, let say a, b,c,d and if the inverse defined by (123 only) so, the order is defined by the number "swap" ? in this case the order is 3, right? not 4 (a,b,c,d), right?

Answer 10

I mean I just swap a,b,c only, d stays. and I have 3 ways to swap it and the result is
c.b.a.d
a,c,b,d
b,c,a,d
after 3 times, it is not back to original form --> not the inverse action, right?

Answer 11

To be an inverse operation, it has to return to the original form with just one operation. It is the inverse of the operation which it undoes.

Answer 12

oooh, but it should be twice, right? 12, swap --->21 , swap again -->12