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? 
> 
> 
> 



Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$?
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! 


