Undecidability of the theory of abelian groups with a subgroup

The theory of abelian groups with an additional predicate denoting a subgroup is undecidable. 0. Introduction. Let L be the first order language with nonlogical symbols 0, +, P, where P is a unary predicate symbol. For any class K of abelian groups let T(K) denote the L-theory of the class of structures , where A E K and B C A is an arbitrary subgroup. Kozlov and Kokorin [4] showed that T(K) is decidable if K is the class of torsion free groups. The main result of this paper is the following: THEOREM 1. Let p be a prime number and let K be the class of abelian groups A such that p9A = 0. Then T(K) is undecidable. An immediate consequence is COROLLARY 2. T(class of all abelian groups) is undecidable. Corollary 2 answers a few questions asked in [4]. 1. Proof of Theorem 1. Let S be a finitely presented semigroup on two generators , a2 and defining relations V,(a1,a2) = W(al,a2)(P of words in a1, a2 an L-sentence 9p such that V = W holds in S if and only if T* F qp. Let A be a p-group and a E A. Put Tr(a) = where h is the p-height, i.e. h(x) = k if and only if x E pKA pKt"A. Put T = , T1 = , T2 = . Note that for each pair , i,j < 2, i #j, T' has a component which is greater than the corresponding component of;T. Forj = 1, 2 let (pj(x,y) be the L-formula x =y= 0 V 3x',y'(P(x')&P(y')&x =p3x'&y ==p3y' & T(X') = Tro & Tr(y') = & h(x' -y') = 1). Let T* be the theory obtained from T(K) by adjoining axioms (i) and (ii) below. Received by the editors March 26, 1975. AMS (MOS) subject classifications (1970). Primary 02G05. 1 Supported by Schweizerischer Nationalfonds. K) American Mathemnatical Society 1976