Asymptotics for the probability that two $n$-cycles generate $S_n$ (or $A_n$). Is there a known asymptotic formula for the probability that two $n$-cycles generate $S_n$ (or $A_n$ in the event that $n$ is odd)?
There seems to be a lot of published research on this question for two arbitrary permutations. This is more a reference request -- no proof necessary. (In fact, I am only interested in the case when $n = p$ is prime, if that is somehow easier or known.)
 A: The case $n=p$ is a prime is particularly easy.
A subgroup of $S_p$ containing a $p$-cycle is either a subgroup of $\mathrm{AGL}(1,p)$, or it is an almost simple $2$-transitive group. (This is due to Burnside.)
Now, these almost simple groups are classified. For most values of $p$, this is only $S_p$ and $A_p$. The exceptional values are $11$, $23$ and when $p$ is of the form $\frac{q^x−1}{q−1}$ for a prime power $q$. (This is due to Guralnick I think.)
So, if $p$ is not exceptional, then the only way for two $p$-cycles not to generate $A_p$ is to be inside a copy of $\mathrm{AGL}(1,p)$. But $\mathrm{AGL}(1,p)$ has a unique (normal) subgroup of order $p$, so the two $p$-cycles must be powers of each other. The probability of this occuring is small and easy to calculate. (Once you pick the first cycle, there are exactly $p−1$ choices for the second one.)
This leaves the exceptional primes, but it could also be handled, as it known what the almost simple group is in this case, and this gives a smallish upper bound on the probability.
