Roots of Identity Matrix

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

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

Given that nn is finite, does integer kk^* exist, such that it is always possible to find a kkk \le k^* for which fk(x)=x,xZnf^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 kk such that Xk=InX^k=I_n has a n×nn \times n matrix solution XX , but, XiInX^i \neq I_n for any i<ki < k

Taking into account that any arbitrary permutations could consist of mm cycles of length i1,i2,,imi_1,i_2,\cdots,i_m with i1+i2++im=ni_1+i_2+\cdots+i_m=n. The problem now boils down to finding the maximum value of the LCM of i1,i2,,imi_1,i_2,\cdots,i_m over all possible values of mm i..e,

k=max\substackm=1ni1+i2++im=nLCM(i1,i2,,im) 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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