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Groups in which every subgroup is nearly permutable

Groups in which every subgroup is nearly permutable Abstract. A relevant theorem of B. H. Neumann states that in a group G each subgroup has finite index in its normal closure if and only if the commutator subgroup G 0 of G is finite, i.e. if and only if G is finite-by-abelian. In this article we prove that in a periodic group G each subgroup has finite index in a permutable subgroup if and only if G contains a finite normal subgroup N such that G=N is a quasihamiltonian group. 2000 Mathematics Subject Classification: 20F24. 1 Introduction A subgroup X of a group G is said to be permutable (or quasinormal) if XH ¼ HX for every subgroup H of G. This concept was introduced by Ore [11], and the behaviour of permutable subgroups was later investigated by several authors. Obviously, normal subgroups are always permutable, and it is easy to show that every maximal permutable subgroup of a group is normal, so that in particular permutable subgroups of finite groups must be subnormal. A group is called quasihamiltonian if all its subgroups are permutable. It has been proved by Stonehewer [14] that permutable subgroups of arbitrary groups are ascendant, so that quasihamiltonian groups are locally http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Groups in which every subgroup is nearly permutable

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References (11)

Publisher
de Gruyter
Copyright
Copyright © 2003 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2003.036
Publisher site
See Article on Publisher Site

Abstract

Abstract. A relevant theorem of B. H. Neumann states that in a group G each subgroup has finite index in its normal closure if and only if the commutator subgroup G 0 of G is finite, i.e. if and only if G is finite-by-abelian. In this article we prove that in a periodic group G each subgroup has finite index in a permutable subgroup if and only if G contains a finite normal subgroup N such that G=N is a quasihamiltonian group. 2000 Mathematics Subject Classification: 20F24. 1 Introduction A subgroup X of a group G is said to be permutable (or quasinormal) if XH ¼ HX for every subgroup H of G. This concept was introduced by Ore [11], and the behaviour of permutable subgroups was later investigated by several authors. Obviously, normal subgroups are always permutable, and it is easy to show that every maximal permutable subgroup of a group is normal, so that in particular permutable subgroups of finite groups must be subnormal. A group is called quasihamiltonian if all its subgroups are permutable. It has been proved by Stonehewer [14] that permutable subgroups of arbitrary groups are ascendant, so that quasihamiltonian groups are locally

Journal

Forum Mathematicum – de Gruyter

Published: Sep 4, 2003

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