ABELIAN P-GROUPS WITH NO INVARIANTS

We prove for abelian p-groups a non-structure theorem relative to appr oximations of Ehrenfeucht-Fraisse games of length omega1 in terms of l inear orderings with no uncountable descending sequences. Our result s hows that there is a group which is too complicated to be characterize d up to isomorphism by the Ehrenfeucht-Fraisse 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 t he Ehrenfeucht-Fraisse game. On the other hand, all the approximations characterize together the notion of isomorphism among groups of cardi nality at most omega1. From the point of view of Stability Theory, our result concerns certain stable theories with NDOP and NOTOP.