Finitary and Artinian-Finitary Groups over the Integers $ℤ$

Abstract

In a series of papers, we have considered finitary (that is, Noetherian-finitary) and Artinian-finitary groups of automorphisms of arbitrary modules over arbitrary rings. The structural conclusions for these two classes of groups are really very similar, especially over commutative rings. The question arises of the extent to which each class is a subclass of the other.

Here we resolve this question by concentrating just on the ground ring of the integers ℤ. We show that even over ℤ neither of these two classes of groups is contained in the other. On the other hand, we show how each group in either class can be built out of groups in the other class. This latter fact helps to explain the structural similarity of the groups in the two classes.