Alternatives for pseudofinite groups

2012/05/16 by Abderezak Ould Houcine, Houcine, Abderezak Ould, Françoise Point +1
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO

paper · pdf · doi:10.48550/arxiv.1205.3533

arxiv created 2012/05/16 · arxiv updated 2012/05/17

Abstract

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 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 rad(G) has a bounded Prüfer rank and the index of the sockel of G/rad(G) is bounded. We show that an ℵ₀-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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