On Zappa's question in the case of alternating groups

In 1962, Guido Zappa asked whether a non-trivial coset of a Sylow $p$-subgroup of a finite group could contain only elements whose orders are powers of $p$. Marston Conder gives a positive answer to this question in the case of $p=5$. It is known that the smallest group satisfying the conditions of this problem must be a non-abelian simple group. In this paper, we prove that the smallest group of the Zappa problem could not be an alternating simple group for any prime $p$.