Abelian \(p\)-groups with no invariants
The main result of this paper is a non-structure theorem for Abelian \(p\)- groups of cardinality \(\aleph_ 1\) and involves the construction of non- isomorphic groups which are similar in a sense defined in terms of game theory. We say that groups \(A\) and \(B\) are \(T\)-equivalent, where \(T\) is a tree of cardinality \(\leq\omega_ 1\) with no uncountable branches, if the isomorphism player has a winning strategy in a certain two-player Ehrenfeucht-Fraïssé game in which the other, non-isomorphism, player has to choose ascending nodes of a branch of \(T\) as he plays. [See \textit{T. Hyttinen} and \textit{H. Tuuri}, Ann. Pure Appl. Logic 52, 203-248 (1991; Zbl 0735.03016), for more on the program.] The main theorem says that, assuming CH, there is an Abelian \(p\)-group \(A\) of cardinality \(\aleph_ 1\) such that for every such tree \(T\), there is a group \(B_ T\) which is \(T\)-equivalent to \(A\), but not isomorphic to \(A\). A consequence, which justifies the term non-structure theorem, is that no sentence of a certain infinitary language, \(M_{\omega_ 2\omega_ 2}\), (whose formulas are syntactical, non-well-founded, trees) can serve as a complete invariant for \(A\). The proof of the main theorem uses an analogous result for linear orders proved by Hyttinen and Tuuri [loc. cit.] together with a Hahn power construction. The result settles a question left open by Hyttinen and Tuuri: a stable theory with NOTOP and NDOP with a strong non-structure result.
scientific article; zbMATH DE number 219101
scientific article