Let $p$ be a prime other than 5 or 7. Are $A\_p$ and $S\_p$ the only subgroups of $S\_p$ that contains a $p$-cycle and a double transposition?


As for $p = 5$, the dihedral group $D\_{10}$ contains a 5-cycle and a double transposition. For $p = 7$, the group $PSL\_3(\mathbb{F}\_2)$ (acting on the projective plane of order 2) contains a 7-cycle and a double transposition.



Transitive subgroup of $S_p$ containing a $p$-cycle and a double transposition
I believe that it was already known to Jordan that if $n \geq 8$ and $G$ is a primitive subgroup of $S\_n$ containing a double transposition, then $G$ contains $A\_n$. As a subgroup of $S\_p$ containing a $p$-cycle is primitive, the answer to your question is ``yes".


