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Jan 2, 2015 at 22:53 comment added Pablo Can you think of a highly transitive action of a finitely generated simple group $G$ with the orbit of every $z \in \mathbb{Z}$ under the action of each $g \in G$ finite? If $G$ is torsion then the last condition is automatic.
Jan 2, 2015 at 21:35 history edited Stefan Kohl CC BY-SA 3.0
Mention that the action is highly transitive.
Jan 2, 2015 at 21:28 history answered Stefan Kohl CC BY-SA 3.0