Abelian p-groups with no invariants

Abstract

We prove for abelian p-groups a non-structure theorem relative to approximations of Ehrenfeucht-Fraïssé games of length ω 1 in terms of linear orderings with no uncountable descending sequences. Our result shows that there is a group which is too complicated to be characterized up to isomorphism by the Ehrenfeucht-Fraïssé game approximated by a fixed ordering. This means that such a group cannot have any complete invariants which are bounded in the sense of these approximations of the Ehrenfeucht-Fraïssé game. On the other hand, all the approximations characterize together the notion of isomorphism among groups of cardinality at most ω 1. From the point of view of Stability Theory, our result concerns certain stable theories with NDOP and NOTOP.