On groups with finitely many derived subgroups.

*(English)*Zbl 1303.20044If \(G\) is a locally graded group for which the set of derived subgroups is finite, it was proved by D. J. S. Robinson and the reviewer [J. Lond. Math. Soc., II. Ser. 71, No. 3, 658-668 (2005; Zbl 1084.20026)] that the derived subgroup \(G'\) of \(G\) is finite. Recall here that a group \(G\) is said to be locally graded if every finitely generated non-trivial subgroup of \(G\) contains a proper subgroup of finite index.

In the paper under review, the author obtains some new informations on the structure of such groups. In fact, he proves that if \(G\) is a locally graded group with only \(k\) derived subgroups (where \(k\) is a positive integer), then the derived subgroup \(G'\) of \(G\) contains a nilpotent-by-Abelian subgroup with index at most \(k!\). Moreover, he shows that any locally graded group with at most \(22\) derived subgroups must be soluble, but the alternating group \(A_5\) has precisely \(23\) derived subgroups.

In the paper under review, the author obtains some new informations on the structure of such groups. In fact, he proves that if \(G\) is a locally graded group with only \(k\) derived subgroups (where \(k\) is a positive integer), then the derived subgroup \(G'\) of \(G\) contains a nilpotent-by-Abelian subgroup with index at most \(k!\). Moreover, he shows that any locally graded group with at most \(22\) derived subgroups must be soluble, but the alternating group \(A_5\) has precisely \(23\) derived subgroups.

Reviewer: Francesco de Giovanni (Napoli)

##### MSC:

20F14 | Derived series, central series, and generalizations for groups |

20E34 | General structure theorems for groups |

20E25 | Local properties of groups |

20E07 | Subgroup theorems; subgroup growth |

##### Keywords:

locally graded groups; numbers of derived subgroups; finite-by-Abelian groups; subgroups of finite index
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\textit{M. Zarrin}, J. Algebra Appl. 13, No. 7, Article ID 1450045, 5 p. (2014; Zbl 1303.20044)

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##### References:

[1] | DOI: 10.1112/S0024610705006484 · Zbl 1084.20026 |

[2] | Dickson L. E., Linear Groups: With an Exposition of the Galois Field Theory (1958) · Zbl 0082.24901 |

[3] | DOI: 10.1090/S0002-9939-1958-0093542-8 |

[4] | DOI: 10.1090/conm/402/07578 |

[5] | DOI: 10.1017/S0017089512000821 · Zbl 1287.20046 |

[6] | DOI: 10.1090/S0002-9904-1968-11953-6 · Zbl 0159.30804 |

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