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 non-empty 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.