On group gradings on PI-algebras

We show that there exists a constant K such that for any PI-algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with [G : U] <= exp(W)(K). A G-grading W = circle plus(g is an element of G) (Wg) is said to be nondegenerate if W-g1 W-g2 ... W-gr not equal 0 for any r >= 1 and any r tuple (g(1), g(2),..., g(r)) in G(r). (C) 2015 Elsevier Inc. All rights reserved.