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The dual structure of almost completely decomposable groups and their endomorphism rings

We consider the so-called almost completely decomposable (acd) groups, which have gained a wide reputation in recent decades that is reflected in Mader’s monograph [1]. By definition, an almost completely decomposable group X is a torsion-free Abelian group of finite rank that contains a completely decomposable subgroup A (a direct sum of groups of rank 1) of finite index. Without loss of generality we may assume that A is the uniquely determined fully invariant completely decomposable subgroup of X of finite index; we call it the regulator of X and write A = R(X). Let A be the class of acd groups having regulator A. It is known that for any group X ∈ A there is a finite set P = P (X) of primes such that X = P p∈P Xp is the sum of fully invariant subgroups Xp ∈ A for which Xp/A is a p-primary finite group. We put E = End X, EA = End A, Ep = End Xp, Hp = Hom(Xp, A), e = exp(X/A) and ep = exp(Xp/A). For each p ∈ P we introduce the natural number b ep = Qq∈P, q 6=p eq and define a group X ′ p = Xp b ep in the divisible hull of Xp such that b epX ′ p = Xp. Clearly Xp ⊂ Xp b ep and b epEp ⊂ Ep.