  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
Pr\"ufer 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.
