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 finitedimensional 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 = Mn(C) we tie up the concepts of braces, grouptheoretic Lagrangians and elementary crossed products. We also manage to compute the intrinsic fundamental group of the diagonal algebras A = C4 and A = C5. (c) 2023 Elsevier Inc. All rights reserved.