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Preprint Number 458

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458. Abderezak Ould Houcine, Françoise Point
Alternatives for pseudofinite groups

Submission date: 16 May 2012.


The famous Tits' alternative states that a linear group either contains a nonabelian free group or is soluble-by-(locally finite). We study in this paper similar alternatives in pseudofinite groups. We show for instance that an \aleph_{0}-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 rad(G) has a bounded Prüfer rank and the index of the sockel of G/rad(G) is bounded. We show that an \aleph_{0}-saturated pseudo-(finite weakly of bounded rank) group either contains a nonabelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between this kind of alternatives and amenability.

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Full text arXiv 1205.3533: pdf, ps.

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