A nilpotency condition for finitely generated soluble groups
- Volume: 9, Issue: 4, page 237-239
- ISSN: 1120-6330
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topDelizia, Costantino. "A nilpotency condition for finitely generated soluble groups." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 9.4 (1998): 237-239. <http://eudml.org/doc/252273>.
@article{Delizia1998,
abstract = {We prove that if \( k > 1 \) is an integer and \( G \) is a finitely generated soluble group such that every infinite set of elements of \( G \) contains a pair which generates a nilpotent subgroup of class at most \( k \), then \( G \) is an extension of a finite group by a torsion-free \( k \)-Engel group. As a corollary, there exists an integer \( n \), depending only on \( k \) and the derived length of \( G \) , such that \( G / Z\_\{n\} (G) \) is finite. For \( k < 4 \), such \( n \) depends only on \( k \).},
author = {Delizia, Costantino},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Commutators; Nilpotency condition; Infinite set; almost nilpotent groups; Engel conditions; finitely generated soluble groups; infinite sets of elements; nilpotent subgroups; extensions; torsionfree Engel groups; derived lengths},
language = {eng},
month = {12},
number = {4},
pages = {237-239},
publisher = {Accademia Nazionale dei Lincei},
title = {A nilpotency condition for finitely generated soluble groups},
url = {http://eudml.org/doc/252273},
volume = {9},
year = {1998},
}
TY - JOUR
AU - Delizia, Costantino
TI - A nilpotency condition for finitely generated soluble groups
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1998/12//
PB - Accademia Nazionale dei Lincei
VL - 9
IS - 4
SP - 237
EP - 239
AB - We prove that if \( k > 1 \) is an integer and \( G \) is a finitely generated soluble group such that every infinite set of elements of \( G \) contains a pair which generates a nilpotent subgroup of class at most \( k \), then \( G \) is an extension of a finite group by a torsion-free \( k \)-Engel group. As a corollary, there exists an integer \( n \), depending only on \( k \) and the derived length of \( G \) , such that \( G / Z_{n} (G) \) is finite. For \( k < 4 \), such \( n \) depends only on \( k \).
LA - eng
KW - Commutators; Nilpotency condition; Infinite set; almost nilpotent groups; Engel conditions; finitely generated soluble groups; infinite sets of elements; nilpotent subgroups; extensions; torsionfree Engel groups; derived lengths
UR - http://eudml.org/doc/252273
ER -
References
top- Delizia, C., Finitely generated soluble groups with a condition on infinite subsets. Istituto Lombardo, Rend. Sc., A 128, 1994, 201-208. Zbl0882.20020MR1433488
- Hall, P., Finite-by-nilpotent groups. Proc. Cambridge Philos. Soc., 52, 1956, 611-616. Zbl0072.25801MR80095
- Heineken, H., Engelsche Elemente der Länge drei. Illinois J. Math., 5, 1961, 681-707. Zbl0232.20073MR131469
- Kappe, L. C. - Kappe, W. P., On three-Engel groups. Bull. Austral. Math. Soc., 7, 1972, 391-405. Zbl0238.20044MR315001
- Lennox, J. C. - Wiegold, J., Extensions of a problem of Paul Erdös on groups. J. Austral. Math. Soc., 21, 1976, 467-472. Zbl0492.20019
- Robinson, D. J. S., Finiteness conditions and generalized soluble groups. Springer-Verlag, Berlin1972. Zbl0243.20033
- Robinson, D. J. S., A course in the theory of groups. Springer-Verlag, Berlin1982. Zbl0836.20001MR648604
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