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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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