  Let $R$ be a finite-dimensional algebra over an algebraically closed field
$F$ graded by an arbitrary group $G$. We prove that $R$ is a graded division
algebra if and only if it is isomorphic to a twisted group algebra of some
finite subgroup of $G$. If the characteristic of $F$ is zero or ${\rm char} F$
does not divide the order of any finite subgroup of $G$ then we prove that $R$
is graded simple if and only if it is a matrix algebra over a
finite-dimensional graded division algebra.
