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Roots of Identity Matrix

I was reading through some problems based on permutations. Particularly, the following problem interested me.

Let \(\mathbb{Z}_n\) denote the set of integers \(\{1,2,\ldots,n\}\) for any positive integer \(n\) and let \(f : \mathbb{Z}_n \rightarrow \mathbb{Z}_n\) be a bijective function defined on \(\mathbb{Z}_n\).

Given that \(n\) is finite, does integer \(k^*\) exist, such that it is always possible to find a \(k \le k^*\) for which \(f^k(x)=x, \forall x \in \mathbb{Z}_n\)?

I suspect that this would turn out to finding the primitive real roots of the identity matrix. That is finding the maximum value of \(k\) such that \(X^k=I_n\) has a \(n \times n\) matrix solution \(X\) , but, \(X^i \neq I_n\) for any \(i < k\)

Taking into account that any arbitrary permutations could consist of \(m\) cycles of length \(i_1,i_2,\cdots,i_m\) with \(i_1+i_2+\cdots+i_m=n\). The problem now boils down to finding the maximum value of the LCM of \(i_1,i_2,\cdots,i_m\) over all possible values of \(m\) i..e,

\[ k^* = \max_{\substack{m=1\cdots n\\ i_1+i_2+\cdots+i_m=n}} LCM (i_1,i_2,\cdots,i_m) \] But, am at a loss in actually proceeding much further.

Note by Janardhanan Sivaramakrishnan
5 months, 2 weeks ago

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