Stonehewer, Stewart E. (Stewart Edward), 1935- and Zacher, G. (Giovanni). (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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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
Date:	August 2003
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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