The **set of all permutations** of an n-element set is denoted by S_{n}.

This is a "group"

(with respect to the composition of maps).

For n > 2, all the groups S_{n} are non-commutative (thus quite complicated).

Note: Every finite group is subgroup of some group S_{n}, thus we see:

the groups S_{n} are in some sense the most complicated ones.

But we have considered only **individual permutations** (and their powers).

The powers of a fixed permutation form a **commutative** subgroup of S_{n}.

(This means: The non-commutativity of the group S_{n} did not play a role.)

What was essential? The possible cycle lengths!

- | The order 8 of the doubling transformation d |

- | Fixed points (thus cycles of length 1) (Secret Santa, Treize). |

- | Cycles of large length (100 prisoners). |

To work with permutations provides many further surprises!

(For example, those of the Catalan combinatorics.)