# Permute till you drop

**Discrete Mathematics**Level 5

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

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

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

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