# Permute till you drop

Let $$\mathbb{Z}_n$$ denote the set of all positive integers less than or equal to an integer $$n$$, and let $$\sigma$$ denote a permutation of $$\mathbb{Z}_n$$.

What is the maximum value of $$k$$ such that there exists a permutation $$\sigma$$ defined on $$\mathbb{Z}_{10}$$, for which $$\sigma^{k}(\mathbb{Z}_{10})=\mathbb{Z}_{10}$$, but $$\sigma^i(\mathbb{Z}_{10})\neq \mathbb{Z}_{10}, \forall 1<i<k$$

Note: Here

1. $$f(S)=S$$ means that $$f(x)=x,\forall x \in S$$.

2. $$f(S) \neq S$$ means that $$f(x)=x$$ is NOT satisfied for at least one element $$x \in S$$.

3. $$\sigma^m(S)=\underset{m\text{ times}}{\underbrace{\sigma \circ \sigma \circ \cdots \circ \sigma}} (S)$$

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