Existentially closed CSA-groups.
A CSA-group (conjugately separated Abelian) is a group \(G\) in which every maximal Abelian subgroup \(H\) is malnormal, that is, \(H\cap H^x=1\) for every \(x\in G\setminus H\). Clearly, every Abelian group is a CSA-group; free groups are examples of non-Abelian CSA-groups. It is known that a non-Abelian CSA-group has no involutions. The class of CSA-groups without involutions is inductive; therefore any member of the class embeds into an existentially closed one. The authors show (answering B.~Poizat's question) that the groups existentially closed in the class of CSA-groups without involutions share many properties of the so called bad groups of finite Morley rank (the existence of which is a major open problem). In particular, they are simple and divisible, their maximal Abelian subgroups are conjugate and cover the group. However, they are not even \(\omega\)-stable. This is proven using the following embedding theorem: every countable CSA-group without involutions embeds into a finitely generated one, which has, up to conjugacy, the same maximal Abelian subgroups, except maybe the infinite cyclic subgroups.
scientific article; zbMATH DE number 2136748
scientific article