Two counterexamples in \(\aleph _ 0\)-categorical groups (Q1061215)

A countable group G is \(\aleph_ 0\)-categorical if the group of automorphisms of G has only finitely many orbits on \(G^ n\) for each \(n\geq 1\). The author exhibits two situations in which \(\aleph_ 0\)- categoricity need not be preserved, in the realms of direct limits and maximal p-subgroups. Theorem A. There is a group \(\Gamma\) and a series of subgroups \(\Gamma_ 1\leq \Gamma_ 2\leq...\leq \Gamma\) such that \(\Gamma =\cup_{n\geq 1}\Gamma_ n\), each \(\Gamma_ n\) is isomorphic to a fixed \(\aleph_ 0\)-categorical group G, but \(\Gamma\) is not \(\aleph_ 0\)- categorical. Theorem B. There is an \(\aleph_ 0\)-categorical group possessing a maximal 2-subgroup which is not \(\aleph_ 0\)-categorical. The author constructs his examples using Boolean powers.