Identities and isomorphisms of finite graded matrix algebras

Abstract

Let $\mathbb{F}$ be a finite field and let $G$ be a group such that every finite subgroup of $G$ is cyclic. In this paper we classify the division gradings by cyclic groups on matrix algebras over $\mathbb{F}$. This yields the classification of the $G$-gradings on these algebras. Moreover if $G$ is abelian we prove that two matrix algebras over $\mathbb{F}$ with $G$-gradings satisfy the same graded identities if and only if they are isomorphic as graded algebras.