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Strongly torsion generated groups. (English) Zbl 0762.20017
A group $$G$$ is torsion generated (t.g.) if it is generated by its elements of finite order and a group $$G$$ is strongly torsion generated (s.t.g.) if for every $$n\geq 2$$ there is an element $$g\in G$$ of order $$n$$ such that the conjugates of $$g$$ generate $$G$$. Examples of s.t.g. groups include the group $$A_ \infty$$ of even finitary permutations of a countable set, the subgroup $$E(R)$$ of the stable general linear group $$GL(R)$$ and the Steinberg groups $$St(R)$$, where $$R$$ is an associative ring with 1.
Now s.t.g. groups were considered by A. J. Berrick [in J. Algebra 139, 190-194 (1991; Zbl 0745.20031)] where he proved that if $$A$$ is an abelian group and $$m\geq 2$$, then there is a s.t.g. group $$G_ m$$ with $$A\simeq H_ m(G_ m,\mathbb{Z})$$ and $$H_ i(G_ m,\mathbb{Z})=0$$ $$(1\leq i < m)$$. Here the authors, by using techniques of G. Baumslag, <span class=”textit”>E</span>. Dyer and C. F. Miller [Topology 22, 27-46 (1983; Zbl 0503.20018)] extend this result and show Theorem 1: Let $$A_ 2,A_ 3,\dots$$ be a sequence of abelian groups. Then there exists a s.t.g. group $$G$$ such that $$H_ n(G,\mathbb{Z})\simeq A_ n$$ for all $$n\geq 2$$. Moreover, if $$\lambda$$ is an infinite cardinal and if each $$A_ \kappa$$ has cardinality $$\leq \lambda$$, then $$G$$ can be chosen to be of cardinality $$\lambda$$ and to have trivial centre. Then by using substantial results from homotopy theory they show Theorem 2: Let $$G$$ be a group having only finitely many non-zero integral homology groups $$H_ n(G,\mathbb{Z})$$. Then any complex linear representation $$\phi: G\to GL_ \kappa(\mathbb{C})$$ is trivial on any finite subgroup of $$G$$. A consequence of Theorem 2 is that when a non-perfect group is generated by torsion elements its integral homology must be non-zero in infinitely many dimensions. Moreover, by Theorem 1 this result is best possible for torsion generated groups.
It was shown by R. G. Swan [in Proc. Am. Math. Soc. 11, 885-887 (1960; Zbl 0096.25302)] that the integral homology of a finite group must be non-zero in infinitely many dimensions. H. Henn obtained [in A note on the homology of locally finite groups (unpublished manuscript)] that if $$G$$ is a locally finite group having only finitely many nonzero homology groups $$H_ n(G,\mathbb{Z})$$, then $$G$$ is acyclic. The authors here obtain more results and we mention Theorem 3. There exists a universal finitely presented acyclic group which is strongly torsion generated.
Reviewer: O.Talelli (Athens)

##### MSC:
 20J05 Homological methods in group theory 20F05 Generators, relations, and presentations of groups 20K40 Homological and categorical methods for abelian groups 20F50 Periodic groups; locally finite groups
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##### References:
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