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**Simple groups with prescribed local properties.**
*(English)*
Zbl 1192.20017

Earlier the authors proved that there exist simple groups that are locally Abelian-by-finite but not locally finite and they have announced that for each \(c\geq 1\) there exist simple groups that are locally (nilpotent of class \(c\))-by-finite but not locally (nilpotent of class \(c-1\))-by-finite.

Here they generalize this by proving the following. Let \(\mathbf V\) be a variety in which the free groups of finite rank are residually finite. Then there exist simple groups \(G\) that are locally \(\mathbf V\)-by-finite and contain a \(\mathbf V\)-free subgroup of countably infinite rank. Further this \(G\) can be chosen to be countable or uncountable.

Equally interestingly the authors also construct here simple periodic groups that are locally residually finite but not locally finite.

Both these theorems are derived from the following. Let \(H\) be an infinite group with a descending series of normal subgroups of finite index whose intersection is \(\langle 1\rangle\). Then there exists a simple group \(S\) such that the following hold. i) The derived subgroup of \(H\) embeds into \(S\). ii) Each finitely generated subgroup of \(S\) is isomorphic to a finite extension of a subgroup of a direct power of \(H\). iii) \(H\) and \(S\) have the same cardinality.

Here they generalize this by proving the following. Let \(\mathbf V\) be a variety in which the free groups of finite rank are residually finite. Then there exist simple groups \(G\) that are locally \(\mathbf V\)-by-finite and contain a \(\mathbf V\)-free subgroup of countably infinite rank. Further this \(G\) can be chosen to be countable or uncountable.

Equally interestingly the authors also construct here simple periodic groups that are locally residually finite but not locally finite.

Both these theorems are derived from the following. Let \(H\) be an infinite group with a descending series of normal subgroups of finite index whose intersection is \(\langle 1\rangle\). Then there exists a simple group \(S\) such that the following hold. i) The derived subgroup of \(H\) embeds into \(S\). ii) Each finitely generated subgroup of \(S\) is isomorphic to a finite extension of a subgroup of a direct power of \(H\). iii) \(H\) and \(S\) have the same cardinality.

Reviewer: B. A. F. Wehrfritz (London)

### MSC:

20E32 | Simple groups |

20E25 | Local properties of groups |

20E26 | Residual properties and generalizations; residually finite groups |

20E10 | Quasivarieties and varieties of groups |

20F50 | Periodic groups; locally finite groups |

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\textit{M. R. Dixon} et al., J. Group Theory 12, No. 5, 745--752 (2009; Zbl 1192.20017)

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### References:

[1] | DOI: 10.1515/JGT.2006.026 · Zbl 1120.20030 · doi:10.1515/JGT.2006.026 |

[2] | Kalunin L., Acta Sci. Math. Szeged 14 pp 69– (1951) |

[3] | Ol’šanski A. Yu., Izv. Akad. Nauk. SSSR Ser. Mat. 44 pp 309– (1980) |

[4] | DOI: 10.1112/blms/9.1.81 · Zbl 0362.20027 · doi:10.1112/blms/9.1.81 |

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