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 $\displaystyle k$-th power of an $\displaystyle n$-cycle $\displaystyle \sigma$ also an $\displaystyle n$-cycle ?
We may assume that $\displaystyle \sigma=(1 2 \cdots n)$.
Consider the "orbit" $\displaystyle \mathcal{O}=\{1,\sigma^k(1),\sigma^{2k}(1),\ldots\ }$ (these are the images of 1 when $\displaystyle \sigma^k$ is repeatedly applied; it may help to draw a sketch in some case, like n=8, k=2 and then k=3).
Saying that $\displaystyle \sigma$ is an $\displaystyle n$-cycle amounts to saying that $\displaystyle |\mathcal{O}|=n$ (this is the cardinality of $\displaystyle \mathcal{O}$).
The cardinality of $\displaystyle \mathcal{O}$ is given by the first $\displaystyle m\geq 1$ such that $\displaystyle \sigma^{km}(1)=1$, which means that $\displaystyle n$ divides $\displaystyle km$. Indeed (in case it is not clear), applying $\displaystyle \sigma$ is like adding 1 modulo $\displaystyle n$; then applying $\displaystyle \sigma^{km}$ is like adding $\displaystyle km$ modulo $\displaystyle n$, and $\displaystyle \sigma^{km}(1)=1$ iff $\displaystyle 1+km = 1$ modulo $\displaystyle n$. We have thus: $\displaystyle |\mathcal{O}|=\min\{m\geq 1|n\mbox{ divides }km\}.$
Now one can easily check that this quantity is equal to $\displaystyle n$ if, and only if $\displaystyle n$ and $\displaystyle k$ are relatively prime. (Or, more precisely $\displaystyle |\mathcal{O}|=\frac{n}{\gcd(n,k)}$).
As a conclusion, the $\displaystyle k$-th power of an $\displaystyle n$-cycle is an $\displaystyle n$-cycle if, and only if $\displaystyle n$ and $\displaystyle k$ are relatively prime.
It may not be the expected method (if $\displaystyle n=12$, you may just look at every case individually), but I found it interesting to look at.
Laurent.