Automorphism groups of countable highly homogeneous partially ordered sets (Q1320983)

We prove Theorem 1. The automorphism group \(G\) of the countable universal poset is simple. Actually, if \(g\in G\) and \(g \neq 1\), then every element of \(G\) can be written in the form \(\prod^ 8_{j = 1}(h^{-1}_{2j-1} gh_{2j-1} \cdot h^{-1}_{2j} g^{-1} h_{2j})\), for some \(h_ 1,\dots,h_{16} \in G\). Theorem 2. If \((\Omega,\leq)\) is a countable highly homogeneous poset, then ``almost all'' finitely generated subgroups of \(\text{Aut}(\Omega,\leq)\) are free. (``Almost all'' is in the Baire Category sense).