Permutations of a set with n elements

A permutation is a bijective map
f : S → S (here, S is a set).
      "bijective map" means: Is s ∈ S, then f(s) ∈ S is defined.
      If f(s) = f(s'), then s = s' (injectivity)
      For every s ∈ S, there is s' ∈ S with f(s') = s (surjectivity).

We denote by I the permutation defined by I(x) = x for all x in S,
(nothing is permuted),
this permutation is called the identity permutation.

Permutations can be composed.

Important: If f is a permutation,
then we are also interested in the powers ft of f,
these are the maps f, f2, f3, ...
(defined by f2(x) = f(f(x)), and f3(x) = f(f(f(x))), and so on,
in addition, we set f0 = I).