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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Solvable groups with $ \pi $-isolators


Authors: A. H. Rhemtulla, A. Weiss and M. Yousif
Journal: Proc. Amer. Math. Soc. 90 (1984), 173-177
MSC: Primary 20F16
DOI: https://doi.org/10.1090/S0002-9939-1984-0727226-2
MathSciNet review: 727226
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Abstract: Let $ \pi $ be any nonempty set of prime numbers. A natural number is a $ \pi $-number precisely if all of its prime factors are in $ \pi $. A group $ G$ is said to have the $ \pi $-isolator property if for every subgroup $ H$ of $ G$, the set $ ^\pi \sqrt H = \{ g \in G;{g^n} \in H$ is a subgroup of $ G$. It is well known that nilpotent groups have the $ \pi $-isolator property for any nonempty set $ \pi $ of primes. Finitely generated solvable linear groups with finite Prüfer rank, and in particular polycyclic groups, have subgroups of finite index with the $ \pi $-isolator property if $ \pi $ is the set of all primes. It is shown here that if $ \pi $ is any finite nonempty set of primes and $ G$ is a finitely generated solvable group, then $ G$ has a subgroup of finite index with the $ \pi $-isolator property if and only if $ G$ is nilpotent-by-finite.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0727226-2
Keywords: Solvable, nilpotent-by-finite, $ \pi $-isolators
Article copyright: © Copyright 1984 American Mathematical Society