Group Gradings on Matrix Algebras

Let $\Phi $ be an algebraically closed field of characteristic zero, $G$ a finite, not necessarily abelian, group. Given a $G$ -grading on the full matrix algebra $A\,=\,{{M}_{n}}\left( \Phi \right)$ , we decompose $A$ as the tensor product of graded subalgebras $A\,=\,B\,\otimes \,C,\,B\,\cong \,{{M}_{p}}\left( \Phi \right)$ being a graded division algebra, while the grading of $C\,\cong \,{{M}_{q}}\left( \Phi \right)$ is determined by that of the vector space ${{\Phi }^{n}}$ . Now the grading of $A$ is recovered from those of $A$ and $B$ using a canonical “induction” procedure.