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Publicly Available Published by De Gruyter July 1, 2013

Alternatives for pseudofinite groups

  • Abderezak Ould Houcine EMAIL logo and Françoise Point
From the journal Journal of Group Theory

Abstract.

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.

Received: 2012-08-29
Published Online: 2013-07-01
Published in Print: 2013-07-01

© 2013 by Walter de Gruyter Berlin Boston

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