Quotient gradings and the intrinsic fundamental group

Quotient grading classes are essential participants in the computation of the intrinsic fundamental group $\pi_1(A)$ of an algebra $A$. In order to study quotient gradings of a finite-dimensional semisimple complex algebra $A$ it is sufficient to understand the quotient gradings of twisted gradings. We establish the graded structure of such quotients using Mackey's obstruction class. Then, for matrix algebras $A=M_n(\mathbb{C})$ we tie up the concepts of braces, group-theoretic Lagrangians and elementary crossed products. We also manage to compute the intrinsic fundamental group of the diagonal algebras $A=\mathbb{C} ^4$ and $A=\mathbb{C} ^5$.