Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-16T19:50:42.262Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Abdollahi, A. and Zarrin, M., ‘Non-nilpotent graph of a group’, Comm. Algebra 38(12) (2010), 4390–4403.CrossRefGoogle Scholar
[2]Barry, M. J. J. and Ward, M. B., ‘Simple groups contain minimal simple groups’, Publ. Mat. 41 (1997), 411–415.CrossRefGoogle Scholar
[3]Blyth, R. D. and Robinson, D. J. S., ‘Insoluble groups with the rewriting property P 8’, J. Pure Appl. Algebra 72 (1991), 251–263.CrossRefGoogle Scholar
[4]Brookes, C. J. B., ‘Engel elements of soluble groups’, Bull. Lond. Math. Soc. 18(1) (1986), 7–10.CrossRefGoogle Scholar
[5]Camp-Mora, S., ‘Locally finite groups with two normalizers’, Comm. Algebra 28 (2000), 5475–5480.CrossRefGoogle Scholar
[6]Endimioni, G., ‘Groupes finis satisfaisant la condition (N,n)’, C. R. Acad. Sci. Paris Ser. I 319 (1994), 1245–1247.Google Scholar
[7] The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.4; 2005, (http://www.gap-system.org).Google Scholar
[8]Huppert, B., Endliche Gruppen, I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[9]Huppert, B. and Blackburn, N., Finite Groups, III (Springer, Berlin, 1982).CrossRefGoogle Scholar
[10]Lennox, J. C. and Wiegold, J., ‘Extensions of a problem of Paul Erdős on groups’, J. Aust. Math. Soc. 31(4) (1981), 459–463.CrossRefGoogle Scholar
[11]Levi, F. W., ‘Groups in which the commutator operation satisfies certain algebraic conditions’, J. Indian Math. Soc. (N.S.) 6 (1942), 87–97.Google Scholar
[12]Polovickiǐ, Y. D., ‘Groups with finite classes of conjugate infinite abelian subgroups’, Soviet Math. (Iz. VUZ) 24 (1980), 52–59.Google Scholar
[13]Pérez-Ramos, M. D., ‘Groups with two normalizers’, Arch. Math. 50 (1988), 199–203.CrossRefGoogle Scholar
[14]Robinson, D. J. S., A Course in the Theory of Groups, 2nd edn (Springer, Berlin–New York, 1995).Google Scholar
[15]Tota, M., ‘Groups with a finite number of normalizer subgroups’, Comm. Algebra 32 (2004), 4667–4674.CrossRefGoogle Scholar
[16]Zarrin, M., ‘Ensuring a group is weakly nilpotent’, Comm. Algebra, to appear.Google Scholar