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[G:U]≤exp⁡(W)K. A G  -grading W=⨁g∈GWgW=⨁g∈GWg is said to be nondegenerate if Wg1Wg2⋯Wgr≠0Wg1Wg2⋯Wgr≠0 for any r≥1r≥1 and any r   tuple (g1,g2,…,gr)(g1,g2,…,gr) in GrGr.