On free subgroups of finite exponent in circle groups of free nilpotent algebras
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 $K$....
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University of Isfahan
2019-06-01
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| Series: | International Journal of Group Theory |
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| Online Access: | http://ijgt.ui.ac.ir/article_22208_1d66e7d97c7526a72c80904843cf42a7.pdf |
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doaj-cd485c8c3d6845e2b2fbd3d266fdc15b2020-11-24T21:21:42ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692019-06-0182294010.22108/ijgt.2017.108014.145522208On free subgroups of finite exponent in circle groups of free nilpotent algebrasJuliane Hansmann0Department of Mathematics University of Kiel, GermanyLet $K$ 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.http://ijgt.ui.ac.ir/article_22208_1d66e7d97c7526a72c80904843cf42a7.pdfgroups of finite exponentrelatively free groupscircle groupfree nilpotent algebraalgebra group |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Juliane Hansmann |
| spellingShingle |
Juliane Hansmann On free subgroups of finite exponent in circle groups of free nilpotent algebras International Journal of Group Theory groups of finite exponent relatively free groups circle group free nilpotent algebra algebra group |
| author_facet |
Juliane Hansmann |
| author_sort |
Juliane Hansmann |
| title |
On free subgroups of finite exponent in circle groups of free nilpotent algebras |
| title_short |
On free subgroups of finite exponent in circle groups of free nilpotent algebras |
| title_full |
On free subgroups of finite exponent in circle groups of free nilpotent algebras |
| title_fullStr |
On free subgroups of finite exponent in circle groups of free nilpotent algebras |
| title_full_unstemmed |
On free subgroups of finite exponent in circle groups of free nilpotent algebras |
| title_sort |
on free subgroups of finite exponent in circle groups of free nilpotent algebras |
| publisher |
University of Isfahan |
| series |
International Journal of Group Theory |
| issn |
2251-7650 2251-7669 |
| publishDate |
2019-06-01 |
| description |
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. |
| topic |
groups of finite exponent relatively free groups circle group free nilpotent algebra algebra group |
| url |
http://ijgt.ui.ac.ir/article_22208_1d66e7d97c7526a72c80904843cf42a7.pdf |
| work_keys_str_mv |
AT julianehansmann onfreesubgroupsoffiniteexponentincirclegroupsoffreenilpotentalgebras |
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1725998743716626432 |