  Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of
Aut($X$), and the field $K$ of $G$-invariant rational functions on $X$. We show
that the subgroup of Aut($X$) that fixes $K$ pointwise is linear algebraic. If
$K$ has transcendence degree $1$ over $k$, then Aut($X$) is an algebraic group.
