Ring of invariants ($R^G=\{r:g(r)=r\text{ for every }g\in G\}$)

For a group acting on a ring by ring automorphisms, the elements fixed by every group element form a subring containing $1$. For finite $G$ acting on a commutative ring, each $r$ satisfies the monic orbit equation ${\prod }_{g\in G}(T-g(r))$, whose coefficients are fixed. Thus $R$ is integral over its invariant subring. This uses an orbit product, not division by the group order.

A fraction of invariant elements is fixed by every automorphism. If that fraction is integral over $R^G$, its monic equation also makes it integral over $R$. A normal domain therefore places it in $R$, and fixedness places it in $R^G$. This proves integral closedness of the invariant subring in its own fraction field. The proof works for any group and in every characteristic; no averaging or finiteness of the group is needed for normality.