1. ## Abstract Help

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.

2. I consider the first question. More generally, when is the $k$-th power of an $n$-cycle $\sigma$ also an $n$-cycle ?

We may assume that $\sigma=(1 2 \cdots n)$.

Consider the "orbit" $\mathcal{O}=\{1,\sigma^k(1),\sigma^{2k}(1),\ldots\ }$ (these are the images of 1 when $\sigma^k$ is repeatedly applied; it may help to draw a sketch in some case, like n=8, k=2 and then k=3).

Saying that $\sigma$ is an $n$-cycle amounts to saying that $|\mathcal{O}|=n$ (this is the cardinality of $\mathcal{O}$).

The cardinality of $\mathcal{O}$ is given by the first $m\geq 1$ such that $\sigma^{km}(1)=1$, which means that $n$ divides $km$. Indeed (in case it is not clear), applying $\sigma$ is like adding 1 modulo $n$; then applying $\sigma^{km}$ is like adding $km$ modulo $n$, and $\sigma^{km}(1)=1$ iff $1+km = 1$ modulo $n$. We have thus: $|\mathcal{O}|=\min\{m\geq 1|n\mbox{ divides }km\}.$
Now one can easily check that this quantity is equal to $n$ if, and only if $n$ and $k$ are relatively prime. (Or, more precisely $|\mathcal{O}|=\frac{n}{\gcd(n,k)}$).

As a conclusion, the $k$-th power of an $n$-cycle is an $n$-cycle if, and only if $n$ and $k$ are relatively prime.

It may not be the expected method (if $n=12$, you may just look at every case individually), but I found it interesting to look at.
Laurent.