Using your Head is Permitted

Let n be an integer greater than 1, and let A, B and C be one to one functions, mapping the elements [0..n-1] to themselves.

All three functions satisfy the property that they are cyclic permutations, in the sense that for any x:0≤x<n the list x, A(x), A(A(x)), A(A(A(x))), etc., goes through all n possible elements before returning to x. This is true not only for A, as in the example, but also for B and C.

The question: for which n do such A, B and C exist, where for all x, B(A(x)) = C(x)?

For each n, either provide a proof of impossibility or an example of such an A-B-C triplet.

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