# 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
3 years, 3 months ago

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