- #1

Halaaku

- 23

- 0

## Homework Statement

Theorem 8.1 of Dan Saracino:

Let f ε S[itex]_{n}[/itex]. Then there exist disjoint cycles f[itex]_{1}[/itex],f[itex]_{2}[/itex]

.. in S such that f= f[itex]_{1}[/itex]°f[itex]_{2}[/itex]...

In proving this theorem, it considers a finite group S_n={1,2,..,n} and chooses x_1 ε S_n. Then it defines x_2= f(x_1), x_3=f(x_2) and so on. The part I do not understand is "because S_n is a finite set, the sequence x_1, x_2 , x_3... must have a repetition. So there must be a first element in the sequence which is the same as the previous element. "

Q: Does a finite set imply that for any sequence of its elements, there HAS to be a repetition?

Q: If so, why should the FIRST element get repeated?

## Homework Equations

Mentioned above.

## The Attempt at a Solution

Trying to understand things...