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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus


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Online
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1435-4446
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Volume 14, Issue 3

Issues

On triple factorizations of finite groups

S. Hassan Alavi / Cheryl E. Praeger
Published Online: 2010-10-13 | DOI: https://doi.org/10.1515/jgt.2010.052

Abstract

Triple factorizations of groups G of the form G = ABA, for proper subgroups A and B, are fundamental in the study of Lie type groups, as well as in geometry. They correspond to flag-transitive point-line incidence geometries in which each pair of points is incident with at least one line. This paper introduces and develops a general framework for studying triple factorizations of this form for finite groups, especially nondegenerate ones where GAB. We identify two necessary and suffcient conditions for subgroups A, B to satisfy G = ABA, in terms of the G-actions on the A-cosets and the B-cosets. This leads to an order (upper) bound for |G| in terms of |A| and |B| which is sharp precisely for the point-line incidence geometries of flag-transitive projective planes. We study in particular the case where G acts imprimitively on the A-cosets, inducing a permutation group that is naturally embedded in a wreath product G 0G 1. This gives rise to triple factorizations for G 0, G 1 and G 0G 1, respectively. We present a rationale for further study of triple factorizations G = ABA in which A is maximal in G, and both A and B are core-free.

About the article

Received: 2009-08-20

Revised: 2010-07-19

Published Online: 2010-10-13

Published in Print: 2011-05-01


Citation Information: Journal of Group Theory, Volume 14, Issue 3, Pages 341–360, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgt.2010.052.

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