Let X be a smooth hypersurface of degree d in the projective space \({\mathbb {P}}^{n+1}\) , and G be an abelian group subgroup of the automorphism group of X where \(n\ge 2\) , \(d\ge 3\) , and \((n,d)\not =(2,4)\) . In this paper, we determine the group structure of G such that the quotient space X/G is smooth. In addition, we show that if X/G is isomorphic to \({\mathbb {P}}^n\) , then X has an outer Galois point p, and the quotient morphism \(q:X\rightarrow X/G\) factors through a projection at an outer Galois point p.
