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On free subgroups of finite exponent in circle groups of free nilpotent algebras | ||
| International Journal of Group Theory | ||
| مقاله 10، دوره 8، شماره 2، شهریور 2019، صفحه 29-40 اصل مقاله (224.12 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2017.108014.1455 | ||
| نویسنده | ||
| Juliane Hansmann* | ||
| Department of Mathematics University of Kiel, Germany | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| groups of finite exponent؛ relatively free groups؛ circle group؛ free nilpotent algebra؛ algebra group | ||
| مراجع | ||
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[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) 55{56. | ||
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