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The Cyclic Notation For Permutation Groups

Consider the set ofelements {1,2,3,,...,n}. We may reorder these, choosing the first in n ways, the second inways and so on. There are n! Possible rearrangements of the set {1,2,3,,...,n}. The set of all possible rearrangements is call the permutation group of orderand labelled

Suppose we have the set {1, 2, 3, 4, 5}.

Possible rearrangements are {2, 3, 1, 5, 4} and {4, 2, 3, 1, 5}.

We may rewrite these rearrangements in cycle form by considering the effect of the rearrangement on each element. Write the original set on one row and the rearrangement below.

The rearrangement sends 1 to 2, 2 to 3 and 3 to 1. This is a closed loop and we can write this as (1 2 3). The arrangement sends 4 to 5 and 5 to 4. This is a closed loop as well and we can write it as (4 5). We can write the permutation in cycle form as (1 2 3)(4 5).

For the second rearrangement we have similarly,

This rearrangement sends 1 to 4, fixes 2 and 3, sends 4 to 1 and fixes 5. 1 and 4 will form a cycle, and 2, 3 and 5 will each form individual cycles. We can write the permutation as (14)(2)(3)(5).

To find the composition of (1 2 3)(4 5) followed by (14)(2)(3)(5), consider the effect of the composition on each of 1, 2, 3, 4, 5.

(1 2 3)(4 5) sends 1 to 2 and (14)(2)(3)(5) fixes 2. The composite will send 1 to 2.

(1 2 3)(4 5) sends 2 to 3 and (14)(2)(3)(5) fixes 3. The composite will send 2 to 3.

(1 2 3)(4 5) sends 3 to 1 and (14)(2)(3)(5) fixes sends 1 to 4. The composite will send 3 to 4.

(1 2 3)(4 5) sends 4 to 5 and (14)(2)(3)(5) fixes 5. The composite will send 4 to 5.

(1 2 3)(4 5) sends 5 to 4 and (14)(2)(3)(5) sends 4 to 1. The composite will send 5 to 1.

The cycle representing the composition of the permutations is (1 2 3 4 5)..

In fact every permutation can be written in cycle form.