Document Zbl 0672.20024 - zbMATH Open

All groups in this review are abelian. Let A be a group. Its localization at the prime p is \(A_ p={\mathbb{Z}}_ p\otimes A\), and its localization at 0 is \(A_ 0={\mathbb{Q}}\otimes A\). A group B is A-like if for all \(p\geq 0\) we have \(A_ p\cong B_ p\). The extended genus of A, EG(A), is the set of isomorphism classes of A-like groups.
For any group X let TX denote the maximal torsion subgroup of X and set \(FX=X/TX\). It is shown (Proposition 1.3) that B is A-like if and only if TB\(\cong TA\) and FB is FA-like. The bulk of the paper is concerned with torsion-free groups, and the case \(A={\mathbb{Z}}^ k\), k a natural number. Let \(A={\mathbb{Z}}^ k\) from now on and consider torsion-free groups only. An A-like group B is represented by a sequence of matrices as follows. Let \(f_ p: B_ p\cong {\mathbb{Z}}^ k_ p\). Then \(f_ 0f_ p^{-1}: {\mathbb{Z}}^ k_ p\to {\mathbb{Q}}^ k\) is a monomorphism which extends uniquely to an element of \(GL_ k({\mathbb{Q}})\). Choosing a fixed basis of \({\mathbb{Z}}^ k\leq {\mathbb{Z}}^ k_ p\leq {\mathbb{Q}}^ k\) one obtains with respect to this basis a sequence \(M_*\) of invertible matrices, the sequential representation of B. If the representations of two A-like groups B and C are the same (or only equivalent in a certain sense) then \(B\cong C\). Not every sequence \(M_*\) of invertible matrices is the sequential representation of an A-like group. However, (Theorem 2.5) \(M_*\) is “realizable” if and only if \(M_ p^{-1}\) has entries in \({\mathbb{Z}}_ p\) for almost all p. There is a more powerful representation: A sequence \(R_*\in \prod_{p}GL_ k({\mathbb{Q}})\) is reduced if \(R_ p\in GL_ k({\mathbb{Z}}[1/p])\) for each prime p and \(R_ p^{-1}\in GL_ k({\mathbb{Z}})\). Every such sequence is realizable, and (Theorem 3.1) every equivalence class of realizable sequences contains a reduced sequence. Given a reduced representation \(R_*\) it is relatively easy to describe the corresponding \({\mathbb{Z}}^ k\)-like group (Theorem 3.3). The rest of the paper contains various examples and applications among these characterizations of \({\mathbb{Z}}^ k\)-like groups which are completely decomposable or almost completely decomposable.

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