Permutation Multiplication and Order of Permutations on 9 Elements

[kevek]

the group of permutations on 9 elements (1,2,3,4,5,6,7,8,9)

Can any I tell me how can I make a multiplication between permutations, and to take some power to permutations?
also, how can I show that determines the order of the permutation.

Many Thanks.

 

[CompuChip]

The "multiplication" in the group is simply performing them after one another.
Take 9 colored balls. Permute the first and third one. Now permute the third and fifth one (note: by third one I mean the one in the third position, not the third ball which is not in the first position).
You have just composed the permutations (13) and (35). In effect, you have moved the first one to the fifth, the fifth one to the third and the third one to the first, so in cycle notation:
(35) o (13) = (153).
(where the composition o is to be read as: "after").

Then as in any group, a power is simply composing the permutation with itself, e.g.
(13)^2 = (13) o (13) = 1
(153)^2 = (153)(153) = (135)
and working out the order is simply composing until you get the identity, e.g.
(153)^3 = (135)(153) = 1

 

[kevek]

Thank you very much for your help.

can you please list some more example in this 9 colored balls please, I am not quite understand the basic thorey inside. I mean (35) after (13) = (351)?

and what is exactly mean of 'identity'