TITLE: Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$?
QUESTION [6 upvotes]: Let $p$ be an odd prime, $g$ a primitive root of $p$ and $A=\{1, 2, \ldots, p-1 \}$.  
Obviously, $\sigma_g(p)=\begin{pmatrix} 
1 & 2 & \ldots & {p-1} \\
g^1\pmod{p} & g^2\pmod{p} & \ldots & g^{p-1}\pmod{p}  
\end{pmatrix}$  is a permutation of $A$.  
I observed that "almost always" $\sigma$ is a product of cycles whose length is strictly less than $p-1$.
If $p=3$ and $g=2$, then $\sigma_2(3)=\begin{pmatrix} 
1 & 2  \\
2 & 1    
\end{pmatrix}=(1\quad2)$ is a cycle of length $3-1=2$.  
If $p=5$ and $g=3$, then $\sigma_3(5)=\begin{pmatrix} 
1 & 2 & 3 & 4   \\
3 & 4 & 2 & 1    
\end{pmatrix}=(1\quad 3\quad 2\quad 4)$ is a cycle of length $5-1=4$.
But these two cases seem to be just exceptions.
For other (small) values of $p$ and every primitive root $g$ of $p$, there are no cycles of length $p-1$. 
I can prove that $g=2$ and $g=\frac{p+1}{2}$ cannot produce such a big cycle, but I do not have any idea how to attack the problem for other values of $g$.
I conjecture that for every $g$ there is not such a big cycle (for sufficiently large $p$)
My question is:  

Do only finitely many primes $p$ exist, such that for some primitive root
  $g$ the permutation  $\sigma_g(p)=\begin{pmatrix}  1 & 2 & \ldots &
{p-1} \\ g^1\pmod{p} & g^2\pmod{p} & \ldots & g^{p-1}\pmod{p}  
 \end{pmatrix}$ is a cycle?

REPLY [3 votes]: As pointed out in the comments, primitive roots with a single cycle appear to be rather common. The standard heuristic argument suggests that there are infinitely many such $p$, and a bit more. 
The share of $N$-cycles among all permutations on $N$ symbols is $1/N$. By the Borel-Cantelli lemma, given an infinite sequence $\{\sigma_i\}$ of independent uniformly distributed random permutations on $N_i$ symbols, if $\sum 1/N_i$ diverges then with probability one infinitely many $\sigma_i$ constitute a single cycle. Since the series of reciprocal primes diverges, we may expect that given a random sequence $\{g_i\}$ of primitive roots mod $p_i$ over all primes, with probability one there are infinitely many $\sigma_i=\sigma_{g_i}(p_i)$ that constitute a single cycle. Of course, since uniform distribution and independence are only heuristic, this is not a proof!