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Hypercentral Groups with all Subgroups Subnormal II

Hypercentral Groups with all Subgroups Subnormal II HYPERCENTRAL GROUPS WITH ALL SUBGROUPS SUBNORMAL II HOWARD SMITH 1. Results An example constructed in [4] shows that there is a group G which is hypercentral of length exactly a>+1 and which has all of its subgroups subnormal. Furthermore, G has rank 2 and is residually finite. Theorem 2 of the same paper states that a group with all subgroups subnormal and having finite abelian subgroup rank and hypercentral length at most co is necessarily nilpotent. Let A^ denote the class of groups in which all subgroups are subnormal and ZA^ the class of hypercentral groups of length at most co. Our first result is: THEOREM 1. A residually finite group in N n ZA^ is nilpotent. Now denote by ft^f,) the class of groups with finite abelian subgroup rank. From [2, Corollary 1 to Theorem 6.36], it is easily seen that locally nilpotent groups with this property are residually of finite rank (/?5 ). A natural generalisation of both [4, Theorem 2] and Theorem 1 above is then provided by: THEOREM 2. A group in the class TV^ n ZA n R% is nilpotent. W r In the course of proving Theorem 2 we shall require the following http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Hypercentral Groups with all Subgroups Subnormal II

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/18.4.343
Publisher site
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Abstract

HYPERCENTRAL GROUPS WITH ALL SUBGROUPS SUBNORMAL II HOWARD SMITH 1. Results An example constructed in [4] shows that there is a group G which is hypercentral of length exactly a>+1 and which has all of its subgroups subnormal. Furthermore, G has rank 2 and is residually finite. Theorem 2 of the same paper states that a group with all subgroups subnormal and having finite abelian subgroup rank and hypercentral length at most co is necessarily nilpotent. Let A^ denote the class of groups in which all subgroups are subnormal and ZA^ the class of hypercentral groups of length at most co. Our first result is: THEOREM 1. A residually finite group in N n ZA^ is nilpotent. Now denote by ft^f,) the class of groups with finite abelian subgroup rank. From [2, Corollary 1 to Theorem 6.36], it is easily seen that locally nilpotent groups with this property are residually of finite rank (/?5 ). A natural generalisation of both [4, Theorem 2] and Theorem 1 above is then provided by: THEOREM 2. A group in the class TV^ n ZA n R% is nilpotent. W r In the course of proving Theorem 2 we shall require the following

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1986

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