ON FREE SUBGROUPS OF FINITE EXPONENT IN CIRCLE GROUPS OF FREE NILPOTENT ALGEBRAS

Abstract. Let K be a commutative ring with identity and N the free nilpotent K-algebra on a nonempty set X. Then N is a group with respect to the circle composition. We prove that the subgroup generated by X is relatively free in a suitable class of groups, depending on the choice of K. Moreover, we get unique representations of the elements in terms of basic commutators. In particular, if K is of characteristic 0 the subgroup generated by X is freely generated by X as a nilpotent group.