Example
The proof actually shows how to find the disjoint cycles decomposition
of a permutation. Consider the permutation (in list notation)
g = [8, 4, 1, 6, 7, 2, 5, 3] in
S8. The following steps lead to the disjoint cycles decomposition.
-
Choose an element in the support of g, for example
1. Now construct the cycle (1, g(1), g2(1), ...).
In this case this cycle is (1, 8, 3). On {1, 3, 8} the permutation
g and the cycle (1, 8, 3) coincide.
-
Next, choose an element
in the support of g, but outside {1, 3, 8}, for example 2.
Construct the cycle (2, g(2), g2(2), ...).
In the case at hand, this cycle is (2, 4, 6).
Then g and (1, 8, 3)(2, 4, 6) coincide on {1, 2, 3, 4, 6, 8}.
-
Choose an element in the support of g but outside {1, 2, 3, 4, 6, 8},
say 5. Construct the cycle
(5, g(5), g2(5), ...), i.e., (5, 7).
Then g and (1, 8, 3)(2, 4, 6)(5, 7) coincide on {1, 2, 3, 4, 5, 6, 7, 8}
and we are done.
Note that the three cycles
(1, 8, 3), (2, 4, 6), (5, 7)
commute, so that g can also be written
as (5, 7)(1, 8, 3)(2, 4, 6) or as (2, 4, 6)(5, 7)(1, 8, 3), etc.