##
**On commutator length of certain classes of solvable groups.**
*(English)*
Zbl 1080.20030

For an element \(g\) in the derived subgroup \(G'\) of a group \(G\), \(c(g)\) denotes the least integer such that \(g\) can be written as a product of \(c(g)\) commutators, where a commutator is an element which can be written as \(a^{-1}b^{-1}ab\) for some \(a,b\in G\). Let \(c(G)=\sup\{c(g)\mid g \in G'\}\). A group \(G\) is called a \(c\)-group if \(c(G)\) is finite.

The main results of the paper under review are: Lemma 1. Let \(G=\langle x_1,\dots,x_n\rangle\) be a free nilpotent-by-Abelian group. Then \(n\leq c(G)\leq n(n+1)/2\).

Theorem 2. Let \(K\) be a nilpotent-by-Abelian normal subgroup with finite index of an \(n\)-generator solvable group \(G\). Then \(c(G)\leq s(s+1)/2+72n^2+47n\), where \(s\) is the number of generators of \(K\).

Corollary 3. Let \(G\) be a solvable group of finite Prüfer rank \(s\). Then \(c(G)\leq s(s+1)/2+72s^2+47s\).

Corollary 4. Let \(A\) be a normal subgroup of a solvable group \(G\) such that \(G/A\) is a \(d\)-generator finite group and \(A\) has finite Prüfer rank \(s\). Then \(c(G)\leq s(s+1)/2+72(s^2+d^2)+47(s+d)\).

In the proof of Theorem 2, the following result of D. Segal [Proc. Lond. Math. Soc., III. Ser. 81, No. 1, 29-54 (2000; Zbl 1030.20017)] is used: In a finite \(d\)-generator solvable group \(G\), every element of \(G'\) can be expressed as a product of \(72d^2+46d\) commutators.

In the statement of Lemma 1, it should be stated \(n\geq 2\) and also the word “free” has no meaning, unless there is a restriction on the nilpotence – i.e., free nilpotent of class at most \(c\) by Abelian.

There is a misprint in the statement as well as the proof of Corollary 4 of the paper (the statement of Corollary 4 which is presented in this review is corrected): \(n\) should read \(d\).

The main results of the paper under review are: Lemma 1. Let \(G=\langle x_1,\dots,x_n\rangle\) be a free nilpotent-by-Abelian group. Then \(n\leq c(G)\leq n(n+1)/2\).

Theorem 2. Let \(K\) be a nilpotent-by-Abelian normal subgroup with finite index of an \(n\)-generator solvable group \(G\). Then \(c(G)\leq s(s+1)/2+72n^2+47n\), where \(s\) is the number of generators of \(K\).

Corollary 3. Let \(G\) be a solvable group of finite Prüfer rank \(s\). Then \(c(G)\leq s(s+1)/2+72s^2+47s\).

Corollary 4. Let \(A\) be a normal subgroup of a solvable group \(G\) such that \(G/A\) is a \(d\)-generator finite group and \(A\) has finite Prüfer rank \(s\). Then \(c(G)\leq s(s+1)/2+72(s^2+d^2)+47(s+d)\).

In the proof of Theorem 2, the following result of D. Segal [Proc. Lond. Math. Soc., III. Ser. 81, No. 1, 29-54 (2000; Zbl 1030.20017)] is used: In a finite \(d\)-generator solvable group \(G\), every element of \(G'\) can be expressed as a product of \(72d^2+46d\) commutators.

In the statement of Lemma 1, it should be stated \(n\geq 2\) and also the word “free” has no meaning, unless there is a restriction on the nilpotence – i.e., free nilpotent of class at most \(c\) by Abelian.

There is a misprint in the statement as well as the proof of Corollary 4 of the paper (the statement of Corollary 4 which is presented in this review is corrected): \(n\) should read \(d\).

Reviewer: Alireza Abdollahi (Isfahan)

### MSC:

20F12 | Commutator calculus |

20F16 | Solvable groups, supersolvable groups |

20F05 | Generators, relations, and presentations of groups |

20F19 | Generalizations of solvable and nilpotent groups |

### Citations:

Zbl 1030.20017
PDF
BibTeX
XML
Cite

\textit{M. Akhavan-Malayeri}, Int. J. Algebra Comput. 15, No. 1, 143--147 (2005; Zbl 1080.20030)

Full Text:
DOI

### References:

[1] | Akhavan-Malayeri M., Houston J. Math. 27 pp 753– |

[2] | DOI: 10.1017/S0017089500032407 · Zbl 0911.20028 |

[3] | Allambergenov Kh. S., Dokl. Akad. Nauk. USSR 4 pp 14– |

[4] | DOI: 10.1016/0019-3577(92)90001-2 · Zbl 0769.20015 |

[5] | DOI: 10.1007/BF01214436 · Zbl 0394.20020 |

[6] | DOI: 10.4153/CJM-1969-126-4 · Zbl 0186.03903 |

[7] | Robinson D. J. S., Finiteness Conditions and Generalized Soluable Groups, Part 2 (1972) |

[8] | Robinson D. J. S., Compositio Math. 31 pp 240– |

[9] | DOI: 10.1112/S002461150001234X · Zbl 1030.20017 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.