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On the Nilpotent Length of Polycyclic Groups

✍ Scribed by Gérard Endimioni

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
146 KB
Volume
203
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let G be a polycyclic group. We prove that if the nilpotent length of each finite quotient of G is bounded by a fixed integer n, then the nilpotent length of G is at most n. The case n s 1 is a well-known result of Hirsch. As a consequence, we obtain that if the nilpotent length of each 2-generator subgroup is at most n, then the nilpotent length of G is at most n. A more precise result in the case n s 2 permits us to prove that if each 3-generator subgroup is abelian-by-nilpotent, then G is abelian-by-nilpotent. Furthermore, we show that the nilpotent length of G equals the nilpotent length of the quotient of G by its Frattini subgroup. ᮊ 1998

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