1. Let σ be the 12-cycle (1 2 3 4 5 6 7 8 9 10 11 12). For which positive integers i is σi also a 12-cycle?
2. If τ = (1 2)(3 4 5) determine whether there is a n-cycle σ (n ≥ 5) with τ = σ^k for
some integer k.
Please help.
I consider the first question. More generally, when is the -th power of an -cycle also an -cycle ?
We may assume that .
Consider the "orbit" (these are the images of 1 when is repeatedly applied; it may help to draw a sketch in some case, like n=8, k=2 and then k=3).
Saying that is an -cycle amounts to saying that (this is the cardinality of ).
The cardinality of is given by the first such that , which means that divides . Indeed (in case it is not clear), applying is like adding 1 modulo ; then applying is like adding modulo , and iff modulo . We have thus:
Now one can easily check that this quantity is equal to if, and only if and are relatively prime. (Or, more precisely ).
As a conclusion, the -th power of an -cycle is an -cycle if, and only if and are relatively prime.
It may not be the expected method (if , you may just look at every case individually), but I found it interesting to look at.
Laurent.