$(2,p)$ generation of alternating group $A_p$

Question

Let $p>7$ be a prime, $a=(1,2,3,...,p)$, $b$ is the product of two arbitrary transpositions such as $b=(2,6)(3,5)$. I found that when $p=11,13,17,19$, the group $\langle a,b \rangle$ is always $A_p$ (use Magma), I want to know whether it is right for all prime numbers $p$. I tried the Jordan's theorem but can't find a short cycle for all situations. Can you prove it or find a counterexample for it? Thanks a lot for any help!

Answer 1

The subgroup $\langle a \rangle$ of $G := \langle a,b \rangle$ is not normal in $G$, so $G$ is not solvable, and hence is doubly transitive by Burnside's Theorem.

Now by Theorem 5.4A of Dixon & Mortimer's book on Permutation Groups, if $G \ne A_p$, then the minimal degree of $G$ (i.e the smallest number of moved points of a non-identity element) is at least $\sqrt{p}$, which is greater than $4$ for $p\ge17$. So $G=A_p$.

The result about minimal degrees of 2-transitive groups is due to Jordan (1871).