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A note on Engel elements in the first Grigorchuk group | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 8، شماره 3، آذر 2019، صفحه 9-14 اصل مقاله (177.96 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2018.109911.1470 | ||
| نویسندگان | ||
| Marialaura Noce1؛ Antonio Tortora* 2 | ||
| 1Department of Mathematics (University of Salerno), Italy - Department of Mathematics and Statistic (University of the Basque Country), Spain | ||
| 2Dipartimento di Matematica, Università di Salerno - Italy | ||
| چکیده | ||
| Let $\Gamma$ be the first Grigorchuk group. According to a result of Bar\-thol\-di, the only left Engel elements of $\Gamma$ are the involutions. This implies that the set of left Engel elements of $\Gamma$ is not a subgroup. The natural question arises whether this is also the case for the sets of bounded left Engel elements, right Engel elements and bounded right Engel elements of $\Gamma$. Motivated by this, we prove that these three subsets of $\Gamma$ coincide with the identity subgroup. | ||
| کلیدواژهها | ||
| Engel elements؛ Grigorchuk group | ||
| مراجع | ||
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[1] A. Ab dollahi, Engel elements in groups , Groups St Andrews 2009 in Bath. 1 , Cambridge Univ. Press, Cambridge, (2011) 94{117. [2] R. Bastos, N. Mansuroglu, A. Tortora and M. Tota, Bounded Engel elements in groups satisfying an identity, Arch. Math. , 110 (2018) 311{318. [3] L. Bartholdi, Algorithmic decidability of Engel's property for automaton groups , Computer science{theory and ap- plications, Springer, (2016) 29{40. [4] L. Bartholdi, Decidability problems in automaton semigroups , arXiv:1705.04598. [5] V. V. Bludov, An example of not Engel group generated by Engel elements , A Conference in Honor of Adalb ert Bovdi's 70th Birthday, Novemb er 18{23, Debrecen, Hungary. [6] P. de la Harp e, Topics in geometric group theory , University of Chicago Press, Chicago, 2000. [7] R. I. Grigorchuk, On Burnside's problem on p erio dic groups, Funktsional. Anal. i Prilozhen. , 14 (1980) 53{54. [8] V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory, The Kourovka Notebook. , arXiv:1401.0300v10. [9] M. No ce, The rst Grigorchuk group , Master's Thesis, Universita di Salerno, 2016. [10] D. J. S. Robinson, A course in the Theory of Groups , 2nd edition, Springer-Verlag, New York, 1996. [11] P. Shumyatsky, A. Tortora and M. Tota, Engel groups with an identity , arXiv:1805.12411. | ||
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