Two theorems on autostability in p-Groups

Abstract

Mal'cev [6] introduced the notion of an autostable structure as a recursive structure where an isomorphism with another recursive structure can be replaced by a recursive isomorphism. Aside from the simplicity and naturalness of this notion, it has fundamental importance since effectiveness questions about algebraic properties are determined in an autostable structure. We will call the problem of characterizing the autostable structures the autostability problem. At present there is no general model theoretic solution to the autostability problem. There are solutions in specific categories. The most general was provided by Nurtazin [8] for the category of decidable models. As a sample application of Nurtazin's Theorem one obtains the following: an algebraically closed field is autostable iff it has finite transcendence degree over its prime field. LaRoche [4] has shown that a Boolean algebra is autostable iff it has only finitely many atoms. Theorem 1 of this paper is classification of the autostable p-groups.

The autostability problem is altered when new relations are added to a structure and solutions to the problem acquire a different flavor. Suppose F is a recursive algebraically closed field and K ⊂ F is an r.e. subfield with F algebraic over K. Now form the recursive structure (F,a)_{a ε K} i.e. we include constants from K in the new structure. The autostability problem for this type of structure is the same as the problem of characterizing those fields K which have a recursively unique algebraic closure. Metakides and Nerode [7] and Smith [11] have shown that this is equivalent to the existence of a splitting algorithm for the separable polynomials over K. In Theorem 2 we classify those p-groups which have a recursively unique divisible closure.

Theorem 2 presupposes that every recursive p-group has a recursive divisible closure. This is proven in Proposition 1. In fact, we show that every recursive abelian group has a recursive divisible closure. We know of no proof in the literature which is sufficiently effective to give this result. The proof given here is drawn from ideas in Rabin [9].

Section 3 of this paper is part of the author's doctoral thesis written under Stephen G. Simpson. Section 4 was motivated by conversations with Anil Nerode.

Supported by an AMS Postdoctoral Research Fellowship and NSF Grant MCS-79-23743.