Finitary groups and Krull dimension over the integers

Abstract

Let $M$ be any Abelian group. We make a detailed study for reasons explained in the Introduction of the normal subgroup $$F_\infty Aut M = \{ g \in Aut M: M(g - 1) is\;a \;minimax\; group\}$$ of the automorphism group $Aut M$ of $M$. The conclusions, although slightly weaker than one would hope, in that they do not fully explain the common behavior of the finitary and the Artinian-finitary subgroups of $Aut M$, are certainly stronger than one might reasonably expect. Our main focus is on residual properties and unipotence.