Let $k$ be a field, $A$ a finitely generated $k$-algebra, put $A^{G}:=\{a \in A \mid g(a)=a ~ \mbox{for all}~g \in G\}$, where $G$ is a finite group of automorphisms of $A$. If
(1) the order of $G$ is not divisible by the characteristic of $k$,
then $A^{G}$ is a finitely generated $k$-algebra.
I saw this statement in I. R. Safarevich's Basic Algebraic Geometry. But I don't know where (1) is used. In in Atiyah Macdonald's Commutative Algebra (exercise 5 of Chapter 7), Condition (1) is omitted.
I wonder what the correct statement is.