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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