摘要 : Let G be a finite group and Da division algebra faithfully G-graded, finite dimensional over its center K, where char(K) = 0. Let e is an element of G denote the identity element and suppose K-0= K boolean AND D-e, the e-center of... 展开 Let G be a finite group and Da division algebra faithfully G-graded, finite dimensional over its center K, where char(K) = 0. Let e is an element of G denote the identity element and suppose K-0= K boolean AND D-e, the e-center of D, contains.nG, a primitive nG-th root of unity, where nGis the exponentof G. To such a G-grading on Dwe associate a normal abelian subgroup Hof G, a positive integer dand an element of Hom(M(H), mu H-n)(G/H). Here mu nHdenotes the group of nH-th roots of unity, n(H)= exp(H), and M(H) is the Schur multiplier of H. The action of G/Hon mu nHis trivial and the action on M(H) is induced by the action of Gon H. Our main theorem is the converse: Given an extension 1 -> H -> G -> Q -> 1, where His abelian, a positive integer d, and an element of Hom(M(H), mu nH)Q, there is a division algebra as above that realizes these data. We apply this result to classify the G-graded simple algebras whose e-center is an closed field of characteristic zero that admit a division algebra form whose e-center contains mu(nG). (C) 2021 Elsevier Inc. All rights reserved. 收起