Quotient gradings and the intrinsic fundamental group

Quotient grading classes are essential participants in the computation of the intrinsic fundamental group π₁(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 group algebra gradings. We establish the graded structure of such quotients using Mackey's obstruction class. Then, for matrix algebras A=M_n(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=C⁴ and A=C⁵.