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**Is there anyone familiar with Fortran programming, who can give me some ideas how can I strart this? Many thanks**

Take 5 colored balls(1,2,3,4,5).

The "multiplication" in the group is simply performing them after one another.

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)

subroutine that determines the order of the permutation

e.g. working out the order is simply composing until you get the identity e.g. (153)^3 = (135)(153) = 1