On Conder’s question on cosets of Sylow subgroups in finite simple groups

In 1962, Guido Zappa asked whether a non-trivial coset of a Sylow 𝑝-subgroup 𝑃 of a finite group could contain only elements whose orders are powers of 𝑝. Marston Conder gave a positive answer to this question for | P | = 5 P=5 . It is known that the smallest group satisfying the conditions of this question must be a non-abelian simple group. He raised a question whether there exists | P | > 5 P>5 that satisfies Zappa’s condition in a simple group. In this paper, we prove that the simple group cannot be an alternating group.