  Let a real Lie group $G$ have a $C^\infty$ action on a real manifold $M$.
Assume every nontrivial element of $G$ has nowhere dense fixpoint set in $M$.
First, we show, in every frame bundle, except possibly the $0$th, that each
stabilizer admits no nontrivial compact subgroups. Second, we show that, if $G$
is connected, then there is a dense open $G$-invariant subset of some higher
order frame bundle of $M$ such that, for any point $x$ in that subset, the
stabilizer in $G$ of $x$ is a discrete, finitely-generated, free-Abelian,
central subgroup of $G$. We derive several corollaries of these two results.
