Finite dimensional graded simple algebras

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 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.