This article is concerned with the p-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime p, the alternating group 프n has a p-basic set. More precisely, we prove that the symmetric group 픖n has a p-basic set with some additional properties, allowing us to deduce a p-basic set for 프n. Our main tool is the concept of generalized perfect isometries introduced by Külshammer, Olsson and Robinson. As a consequence we obtain some results on the decomposition numbers of 프n.