Totally Crawley groups

Abstract

All groups are abelian p-groups, for a fixed prime p. In an important paper [Meg2], C. Megibben proved that it is undecidable in ordinary set theory (ZFC) whether or not every (o-)Crawley group of cardinality 8, is C-cyclic. Specifically, he showed that the answer is “yes” assuming V= L, and the answer is “no” assuming MA + 1 CH. Recently Mekler and Shelah showed that, assuming V= L, Crawley groups of arbitrary cardinality are L-cyclic ([MSl], [MS2]). In this paper we investigate the question of generalizing these results by replacing u by an arbitrary countable limit ordinal 1. As we shall see, the generalization is by no means routine, and, in fact, some questions remain open. Recall that a group G is called /2-Zippin or uniquely I-elongating if l(G), the length of G, is i and whenever A and B are &elongations of G (i.e., A/p’A z Gz B/p”B), then every isomorphism between piA and pLB extends to an isomorphism between A and B. Call G i-Crawley if whenever A and B are &elongations of G by Z(p) (so, in addition, p”A z Z(p) g p’B), then A and B are isomorphic (or, equivalently, every