Gradings on Matrix Algebras of Prime Order

We give a classification of the group gradings on \(M_p(K)\) $M_p(K)$ , here p is a prime number and K is a field that contains a primitive p-th root of unity if \(\textrm{char}\, K\ne p\) $\text{char}K\ne p$ . The gradings are isomorphic to an elementary grading, to a division grading with support isomorphic to \({\mathbb {Z}}_p\) ${\mathbb{Z}}_p$ or to a division grading with support isomorphic to \({\mathbb {Z}}_p\times {\mathbb {Z}}_p\) ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$ .