Let $G$ be a degree $n$ group, i.e., a subgroup of the symmetric group $S_n$. Let $p(G)$ be the number of $n$-cycles in $G$ divided by the size of $G$.
- If $G$ is a cyclic transitive group, then $p=\varphi(n)/n$.
- If $G=S_n$, then $p=1/n$.
- (If $G$ is not transitive, then $p=0$)
The question is whether $p(G)\leq \varphi(n)/n$ for every degree $n$ group?
- One can see that $p(G)=k/n$, where $k$ is the number of conjugacy classes of $n$-cycles, so the answer is YES if $n$ is prime.
- Numerical testing shows the answer is YES for $n\leq 30$ and for primitive groups for $n\leq 1000$.
- There are non-cyclic groups achieving the bound $\varphi(n)/n$, e.g., the wreath product of cyclic groups.
Edit: Recently Joachim König solved this using the classification both in the induction basis as Michael Giudici mentioned and also in the induction step. I guess we should wait for the paper which is now in refereeing process.