There is a known result says that for $G$ as a finite group. If $A$ be an (assocatively) $G$-graded associative algebra such that the homogeneous component $A_1$ satisfies a polynomial identity of degree $d$, then the entire algebra $A$ satisfies a polynomial identity with degree bounded above by an explicit function of $d$ and $|G|$. My question is about what is the meaning of
associatively $G$-graded
in this context?