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Hypercentral Groups with all Subgroups Subnormal

Hypercentral Groups with all Subgroups Subnormal HYPERCENTRAL GROUPS WITH ALL SUBGROUPS SUBNORMAL HOWARD SMITH 1. Results In [2], Heineken and Mohamed construct a metabelian group G with the property that every proper subgroup of G is nilpotent and subnormal in G but G itself has trivial centre. Papers [1], [3] and [6] each show how 2*° pairwise non-isomorphic groups may be constructed, each with properties which are similar to those possessed by the group G. On the other hand, it is shown in [2] that a hypercentral group having all proper subgroups nilpotent and subnormal is in fact nilpotent. The main purpose of this note is to show that the condition "all proper subgroups nilpotent" is necessary here, i.e. that there is a non-nilpotent, hypercentral group having all its subgroups subnormal, thus answering a question posed in [4] . Two further questions, in related areas, are also answered. The first of these is concerned with the series of "ascending normalisers" of a subgroup H of a group G, given by N^H) = N {H) and, for i = 1,2,..., N (H) = N (Nj(H)). A group G is G i + l c said to satisfy the normaliser condition (NC) if every proper subgroup of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Hypercentral Groups with all Subgroups Subnormal

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/15.3.229
Publisher site
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Abstract

HYPERCENTRAL GROUPS WITH ALL SUBGROUPS SUBNORMAL HOWARD SMITH 1. Results In [2], Heineken and Mohamed construct a metabelian group G with the property that every proper subgroup of G is nilpotent and subnormal in G but G itself has trivial centre. Papers [1], [3] and [6] each show how 2*° pairwise non-isomorphic groups may be constructed, each with properties which are similar to those possessed by the group G. On the other hand, it is shown in [2] that a hypercentral group having all proper subgroups nilpotent and subnormal is in fact nilpotent. The main purpose of this note is to show that the condition "all proper subgroups nilpotent" is necessary here, i.e. that there is a non-nilpotent, hypercentral group having all its subgroups subnormal, thus answering a question posed in [4] . Two further questions, in related areas, are also answered. The first of these is concerned with the series of "ascending normalisers" of a subgroup H of a group G, given by N^H) = N {H) and, for i = 1,2,..., N (H) = N (Nj(H)). A group G is G i + l c said to satisfy the normaliser condition (NC) if every proper subgroup of

Journal

Bulletin of the London Mathematical Society – Wiley

Published: May 1, 1983

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