LAUSR

Given integers $d\ge 3$ and $N\ge 2$. Let $G$ be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree $d$ in the complex projective space $\mathbb{P}^{N-1}$. Suppose $G\subset PGL(N, \mathbb{C})$ can be lifted to a subgroup of $GL(N, \mathbb{C})$. Suppose more that there exists an element $g$ in $G$ with order a prime-power, such that $G/\langle g\rangle$ has order coprime to $d-1$. Then all possible $G$ are determined. As an application, we derive all possible orders of automorphisms of smooth hypersurfaces for any given $(d, N)$.