- #1
Halaaku
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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...