Graded Identities and Induced Gradings on Group Algebras

Abstract

The group algebra FG of the group G has the natural G-grading. It can be also graded by G / H for any normal subgroup H where the homogeneous components are the span of cosets. We show that if FG = R =⊕ _s∈S R _s is an S-grading such that any g ∈ G ⊂ FG is homogeneous in S-grading then S contains a subgroup G / H and it is actually G / H-grading where all S-homogeneous components are cosets of H. For a finitely generated normal subgroup H we also prove that FG satisfies a G / H-graded identity if and only if F[G / H] is a P.I. algebra.

Research supported by NSERC grant A-5300

Research supported by RFBR grants 02-01-00219 and 00-15-96128