**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** f^{t} of f,

these are the maps f, f^{2}, f^{3}, ...

(defined by f^{2}(x) = f(f(x)), and f^{3}(x) = f(f(f(x))), and so on,

in addition, we set f^{0} = I).