Main Theorem. If f is a permutation of a finite set, then there is a natural number t with ft = I.

More precisely: If n is a natural number, let n! = 1×2×3×...×n,
(n! is called "n factorial"),

for example:	4! = 1×2×3×4 = 24,
and	5! = 1×2×3×4×5 = 120.

Main Theorem. If f is a permutation of a set of n elements, then there is a natural number t ≤ n! with ft = I.

The smallest such number t is called the order ord(f) of f.

Reformulation:

Every permutation f of a finite set S has an order ord(f). If S is a set with n elements, then ord(f) ≤ n!

The proof is not difficult! Let us have a look at the proof.