Strongly torsion generated groups.

*(English)*Zbl 0762.20017A 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 H. Henn obtained [in A note on the homology of locally finite groups (unpublished manuscript)] that if \(G\) is a locally finite group having only finitely many nonzero homology groups \(H_ n(G,\mathbb{Z})\), then \(G\) is acyclic.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.

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 |

##### Keywords:

elements of finite order; s.t.g. groups; integral homology groups; non-perfect groups; generated by torsion elements; torsion generated groups; locally finite groups; universal finitely presented acyclic group; strongly torsion generated
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\textit{A. J. Berrick} and \textit{C. F. Miller III}, Math. Proc. Camb. Philos. Soc. 111, No. 2, 219--229 (1992; Zbl 0762.20017)

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##### References:

[1] | Baumslag, Bull. London Math. Soc. 20 pp 239– (1988) |

[2] | Berrick, An Approach to Algebraic (1982) |

[3] | Baumslag, J. Pure Appl. Algebra 16 pp 1– (1980) |

[4] | Atiyah, Inst. Hautes tudes Sci. Publ. Math. 9 pp 23– (1961) |

[5] | Swan, Proc. Amer. Math. Soc. 11 pp 885– (1960) |

[6] | Milnor, Introduction to Algebraic K-Theory. Ann. of Math. Studies (1971) · Zbl 0237.18005 |

[7] | Miller, Ann. of Math. 120 pp 39– (1984) |

[8] | Miller, Proceedings of the Workshop on Algorithms, Word Problems and Classification in Combinatorial Group Theory (1990) |

[9] | Malcev, Mat. Sb. 8 pp 405– (1940) |

[10] | Lyndon, Combinatorial Group Theory (1977) · doi:10.1007/978-3-642-61896-3 |

[11] | Berrick, Groups and Classifying Spaces |

[12] | Berrick, J. Algebra 139 pp 190– (1991) |

[13] | Berrick, Bull. London Math. Soc. 22 pp 227– (1990) |

[14] | Berrick, Enseign. Math. 31 pp 151– (1985) |

[15] | Baumslag, Topology 22 pp 27– (1983) |

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