order of permutations

Question

Dont know how to do the permutations

E.g (123)(12). If we do the permutation from the left first:

(123)(12)=(1)(23)=(23)

(1234)^2=(1234) (1234)=(13)(24)

What is the concept behind it???

Answer 1

Favorite Answer

You can think them as the composition of two fn.(except the order of operation).

(12): 1 -> 2, 2 -> 1

(123): 1 -> 2, 2->3 , 3->1

(123)(12): 1->2->1, 2->3, 3->1->2, so we obtain (1)(23) or (23)

(1234)^2=(1234)(1234)

1->2->3

2->3->4

3->4->1

4->1->2

so (1234)^2=(13)(24)

2010-10-11 18:16:01 補充：

1.

Let f=(12), g=(123), then f。g=(123)(12).

f。g: 1 -> f(2)=1

f。g: 2 -> f(3)=3

f。g: 3 -> f(1)=2

so (123)(12)=f。g=(1)(23)=(23)

2.

f=(1234), (1234)^2=f。f

f。f(1)=f(2)=3

f。f(2)=f(3)=4

f。f(3)=f(4)=1

f。f(4)=f(1)=2

so (1234)^2=f。f=(13)(24)

Answer 2

Why we close the brakcet at 3?

It is my thought: (1234)(1234)

First bracket: 1->2, 2 in the first bracket ->3 in the second bracket. Why we closed bracket here? Why dont we keep on the rotation, like 3 in the second bracket -> 4 in the first bracket and 4 in the first bracket -> 1.