Is $\varphi(n)/n$ the maximal portion of $n$-cycles in a degree $n$ group?

Question

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 $G$.

Examples:

1. If $G$ is a cyclic transitive group, then $p=\varphi(n)/n$.
2. If $G=S_n$, then $p=1/n$.
3. (If $G$ is not transitive, then $p=0$)

The question is whether $p(G)\leq \varphi(n)/n$ for every degree $n$ group?

Note:

1. 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.
2. Numerical testing shows the answer is YES for $n\leq 30$ and for primitive groups for $n\leq 1000$.
3. 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.

Answer 1

It is true for all primitive groups: The primitive groups of degree n containing an n-cycle were independently classified in

Li, Cai Heng The finite primitive permutation groups containing an abelian regular subgroup. Proc. London Math. Soc. (3) 87 (2003), no. 3, 725--747. )

and

Jones, Gareth A. Cyclic regular subgroups of primitive permutation groups. J. Group Theory 5 (2002), no. 4, 403--407.

They are the groups G such that

-$C_p\leqslant G\leqslant AGL(1,p)$ for p a prime

-$A_n$ for n odd, or $S_n$

-$PGL(d,q)\leqslant G \leqslant P\Gamma L(d,q)$: here there is a unique class of cyclic subgroups generated by an n-cycle except for $G=P\Gamma L(2,8)$ in which case there are two.

-$(G,n)=(PSL(2,11),11), (M_{11},11), (M_{23},23)$

All these groups satisfy the bound.

Gordon Royle has pointed out to me that the bound does not hold for elements of order n. The smallest examples which do not meet the bound are of degree 12 and are the groups numbered 263 and 298 in the catalogue of groups of degree 12 in Magma.

Answer 2

This answer exists to record a false approach I had.

Let $G$ be a subgroup of $S_n$. Call a subgroup of $G$ "maximal cyclic" if it is generated by an $n$-cycle. False Statement: Any two maximally cyclic subgroups are conjugate.

This is true in a number of cases, and implies the claimed result. However, as Dmitri led me to realize, it is false for the alternating group $A_9$.

I used to have a longer answer explaining this. I deleted it and replaced it with this answer because it got an upvote which, as I understand it, removed this excellent question from the unanswered list. So PLEASE DON'T VOTE THIS UP! Let's see if someone can come up with a real answer!