On free subgroups of finite exponent in circle groups of free nilpotent algebras

Document Type : Research Paper


Department of Mathematics University of Kiel, Germany


‎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.


Main Subjects

[1] T. Andreescu and D. Andrica, Number Theory , Birkhauser Boston, Inc., Boston, MA, 2009.
[2] M. Hall, The Theory of Groups , Chelsea Publishing Co., New York, 1976.
[3] M. Lothaire, Combinatorics on Words , Cambridge University Press, Cambridge, 1997
[4] W. Magnus, Beziehungen zwischen Grupp en und Idealen in einem sp eziellen Ring, Math. Ann. , 111 (1935) 259{280.
[5] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory , Dover Publications, Inc., New York, 1976.
[6] R. Quintana Jr., A b ound on the order of nitely generated nilp otent groups with an exp onent, J. Algebra , 127 (1989)