Group identities on units of group algebras

Let U, be the roup of units of the group algebra FG of a group G over a fie ld F. Suppose that either F is infinite or G has an element of infinite ord er. We characterize groups G so that U satisfies a group identity. Under th e assumption that G module the torsion elements is nilpotent this gives a c omplete classification of such groups. For torsion groups this problem has already been settled in recent years. (C) 2000 Academic Press.