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ON GROUPS WITH A FINITE NUMBER OF NORMALISERS

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  14 February 2012

MOHAMMAD ZARRIN
MOHAMMAD ZARRIN*
Affiliation:
Department of Mathematics, University of Kurdistan, PO Box 416, Sanandaj, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box: 19395-5746, Tehran, Iran (email: m.zarrin@uok.ac.ir, zarrin78@gmail.com)
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Abstract

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Groups having exactly one normaliser are well known. They are the Dedekind groups. All finite groups having exactly two normalisers were classified by Pérez-Ramos [‘Groups with two normalizers’, Arch. Math.50 (1988), 199–203], and Camp-Mora [‘Locally finite groups with two normalizers’, Comm. Algebra28 (2000), 5475–5480] generalised that result to locally finite groups. Then Tota [‘Groups with a finite number of normalizer subgroups’, Comm. Algebra32 (2004), 4667–4674] investigated properties (such as solubility) of arbitrary groups with two, three and four normalisers. In this paper we prove that every finite group with at most 20 normalisers is soluble. Also we characterise all nonabelian simple (not necessarily finite) groups with at most 57 normalisers.

Keywords

normaliser subgroupssimple groupsn-Engel elementssoluble groups

MSC classification

Secondary: 20E34: General structure theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 416 - 423
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

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