Main Theorem.
If f is a permutation of a finite set,

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,

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).

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