= Normality of a ring of invariants
{title2=$R\text{ normal domain}\Longrightarrow R^G\text{ normal domain}$}
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.
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