In this paper, we determine the automorphism group of the p-cones (p\neq 2) in dimension greater than two. In particular, we show that the automorphism group of those p-cones are the positive scalar multiples of the generalized permutation matrices that fix the main axis of the cone. Next, we take a look at a problem related to the duality theory of the p-cones. Under the Euclidean inner product it is well-known that a p-cone is self-dual only when p=2. However, it was not known whether it is possible to construct an inner product depending on p which makes the p-cone self-dual. Our results shows that no matter which inner product is considered, a p-cone will never become self-dual unless p=2 or the dimension is less than three.