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Some finite solvable groups with non-trivial lattice endomorphisms

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Stonehewer, Stewart E. and Zacher, G.. (2003) Some finite solvable groups with non-trivial lattice endomorphisms. Bulletin of the Australian Mathematical Society , Vol.68 (No.1). pp. 141-153. ISSN 0004-9727

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Official URL: http://dx.doi.org/10.1017/S0004972700037497

Abstract

The main purpose of this paper is to exhibit a doubly-infinite family of examples which are extensions of a p-group by a p′-group, with the action satisfying some conditions of Zappa (1951), arising from his study of dual-standard (meet-distributive) subgroups. The examples show that Zappa's conditions do not bound the nilpotency class (or even the derived length) of the p-group. The key to this work is found in closely related conditions of Hartley (published here for the first time). The examples use some exceptional relationships between primes.

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Mathematics

Library of Congress Subject Headings (LCSH):

Lattice theory, Finite groups, Endomorphisms (Group theory), Group theory

Journal or Publication Title:

Bulletin of the Australian Mathematical Society

Publisher:

Cambridge University Press

ISSN:

0004-9727

Official Date:

August 2003

Dates:

Date	Event
August 2003	Published

Volume:

Vol.68

Number:

No.1

Page Range:

pp. 141-153

Identification Number:

10.1017/S0004972700037497

Status:

Peer Reviewed

Access rights to Published version:

Open Access

References:

[1] Ft. Crandall, C. Pomerance, Prime numbers, a computational perspective (Springer-Verlag, Berlin, 2001).
[2] J. Dixon, M. du Sautoy, A. Mann and D. Segal, Analytic pro-p-groups, London Mathematical Society Lecture Notes Series 157 (Cambridge University Press, Cambridge, 1991).
[3] K. Doerk and T.O. Hawkes, Finite soluble groups (de Gruyter, Berlin, New York, 1992).
[4] B. Huppert, Endliche Gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967).
[5] B. Huppert and N. Blackburn, Finite groups II (Springer-Verlag, Berlin, Heidelberg, New York, 1982).
[6] A. Lubotzky and A. Mann, 'Powerful p-groups I, finite groups', J. Algebra 105 (1987), 484-505.
[7] S. Mattarei, Retrieving information about a group from its character table, (Ph.D. Thesis) (University of Warwick, Warwick, 1992).
[8] H. Neumann, Varieties of groups (Springer-Verlag, Berlin, Heidelberg, New York, 1967).
[9] R. Schmidt, Subgroup lattices of groups, de Gruyter Expositions in Mathematics 14 (de Gruyter, Berlin, New York, 1994).
[10] S.E. Stonehewer, G. Zacher, 'Dual-standard subgroups of finite and locally finite groups', Manuscripta Math. 70 (1991), 115-132.
[11] M. Suzuki, Structure of a group and the structure of its lattice of subgroups (Springer-Verlag, Berlin, Gottingen, Heidelberg, 1967).
[12] G. Zappa, 'Sulla condizione perche un emitrofismo inferiore tipico tra due gruppi sia un emotropismo', Giorn. Mat. Battaglini 80 (1951), 80-101.

URI:

http://wrap.warwick.ac.uk/id/eprint/773

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