Division algebras graded by a finite group

Let G be a finite group and D division algebra faithfully -graded, dimensional over its center K , where c h r ( ) = 0 . e ∈ denote the identity element suppose ∩ -center of contains ζ n primitive -th root unity, is exponent To such -grading on we associate normal abelian subgroup H positive integer d an o m M μ / Here denotes roots x p Schur multiplier The action trivial induced by Our main theorem converse: Given extension 1 → Q abelian, there as above that realizes these data. We apply this result to classify -graded simple algebras whose algebraically closed field characteristic zero admit form