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Torsion generators for all abelian groups. (English) Zbl 0745.20031
A group \(G\) is called a perfect group when \(G\) coincides with its commutator subgroup. Pursueing his former research on embedding any abelian group onto the center of a perfect group [J. Pure Appl. Algebra 44, 35-43 (1987; Zbl 0614.20038)], the author proves the following two theorems: 1. Let \(A\) be any abelian group. Then there exists a perfect group \(G\) such that \(A\) is isomorphic to the center of \(G\), and that, for any \(n\geq 2\), \(G\) is normally generated by a single element of order \(n\). 2. Let \(A\) be any abelian group. Then, for any \(m\geq 2\), there exists a group \(G_ m\) such that \(A\cong H_ m(G_ m)\) and \(H_ i(G_ m)=0\) \((1\leq i< m)\), where, for any \(n\geq 2\), \(G_ m\) is normally generated by a single element of order \(n\). (The homology is taken with trivial integer coefficients).
Reviewer: K.-y.Honda (Tokyo)

MSC:
20E22 Extensions, wreath products, and other compositions of groups
20J05 Homological methods in group theory
20K40 Homological and categorical methods for abelian groups
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups
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References:
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