  We provide results on the smoothness of normalisers in connected reductive
algebraic groups $G$ over fields $k$ of positive characteristic $p$.
Specifically we we give bounds on $p$ which guarantee that normalisers of
subalgebras of $\mathfrak{g}$ in $G$ are smooth, i.e.\ so that the Lie algebras
of these normalisers coincide with the infinitesimal normalisers. One of our
main tools is to exploit cohomology vanishing of small dimensional modules.
Along the way, we obtain complete reducibility results for small dimensional
modules in the spirit of similar results due to Jantzen, Guralnick, Serre and
Bendel--Nakano--Pillen.
