  Let $\mathbb{k}$ be an algebraically closed field of characteristic zero. Let
$D$ be a division algebra of degree $d$ over its center $Z(D)$. Assume that
$\mathbb{k}\subset Z(D)$. We show that a finite group $G$ faithfully grades $D$
if and only if $G$ contains a normal abelian subgroup of index dividing $d$. We
also prove that if a finite dimensional Hopf algebra coacts on $D$ defining a
Hopf-Galois extension, then its PI degree is at most $d^2$. Finally, we
construct Hopf-Galois actions on division algebras of twisted group algebras
attached to bijective cocycles.
