Maps to itself... eventually

Algebra Level 3

Let f:NNf : \mathbb N \to \mathbb N denote a function such that the table below is fulfilled.

xx123456789101112
f(x)f(x) 810125211971436

Which of the following is true for integer 1S121\leq S \leq 12 ?

Bonus: Show that for any such permutation σ\sigma, there is an nn such that σn\sigma ^ n is the identity permutation.

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