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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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Volume 16, Issue 4


Alternatives for pseudofinite groups

Abderezak Ould Houcine
  • Département de mathématique, Le Pentagone, Université de Mons, 20, place du Parc, B-7000 Mons, Belgium; and Institut Camille Jordan, Bâtiment Braconnier (Site de la Doua), Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
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Published Online: 2013-07-01 | DOI: https://doi.org/10.1515/jgt-2013-0006


The famous Tits alternative states that a linear group either contains a non-abelian free group or is soluble-by-(locally finite). In this paper we study similar alternatives in pseudofinite groups. We show, for instance, that an ℵ0-saturated pseudofinite group either contains a subsemigroup of rank 2 or is nilpotent-by-(uniformly locally finite). We call a class of finite groups G weakly of bounded rank if the radical has a bounded Prüfer rank and the index of the socle of is bounded. We show that an ℵ0-saturated pseudo-(finite weakly of bounded rank) group either contains a non-abelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between these kind of alternatives and amenability.

About the article

Received: 2012-08-29

Published Online: 2013-07-01

Published in Print: 2013-07-01

Citation Information: Journal of Group Theory, Volume 16, Issue 4, Pages 461–495, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgt-2013-0006.

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