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**Commutator length of Abelian-by-nilpotent groups.**
*(English)*
Zbl 0911.20028

Let \(c(G)\) denote the minimal number such that every element of the derived subgroup \(G'\) of a group \(G\) can be expressed as a product of at most \(c(G)\) commutators, or \(c(G)=\infty\) in the case of unbounded expressability. A number of known theorems state finiteness or infiniteness of \(c(G)\). A very few of them give exact values of \(c(G)\).

Let \(M_{nt}\) denote the free metabelian nilpotent group of rank \(n\) and class \(t\), \(M_n\) denotes the free metabelian group of rank \(n\). Kh. S. Allambergenov and the reviewer proved [in Dokl. Akad. Nauk UzSSR 1984, No. 4, 14-15 (1984; Zbl 0578.20025), full proofs in VINITI 1985, No. 9566-85, 19 p.] that \(c(M_{n2})=[n/2]\) for every \(n\geq 2\), \(c(M_{n3})=n\) for every \(n\geq 3\), and \(c(M_{nt})=n\) for every \(n\geq 2\), \(t\geq 4\). Kh. S. Allambergenov proved in his Thesis (Omsk, 1985), that \(c(M_n)=n\) for every \(n\geq 2\).

The authors prove that \(c(M_{23})=2\) completing the set of results of this kind for the class of groups \(\{M_{nt}\}\). It follows that the same assertions are valid for the class \(N_{nt}\), where \(N_{nt}\) denotes the free nilpotent group of rank \(n\) and class \(t\).

The authors also prove that every element of the derived subgroup \(G'\) of a free abelian-by-nilpotent group \(G\) of rank \(n\) can be expressed as a product of \(n\) commutators. Thus by Allambergenov-Roman’kov’s result cited above \(c(G)=n\).

Let \(M_{nt}\) denote the free metabelian nilpotent group of rank \(n\) and class \(t\), \(M_n\) denotes the free metabelian group of rank \(n\). Kh. S. Allambergenov and the reviewer proved [in Dokl. Akad. Nauk UzSSR 1984, No. 4, 14-15 (1984; Zbl 0578.20025), full proofs in VINITI 1985, No. 9566-85, 19 p.] that \(c(M_{n2})=[n/2]\) for every \(n\geq 2\), \(c(M_{n3})=n\) for every \(n\geq 3\), and \(c(M_{nt})=n\) for every \(n\geq 2\), \(t\geq 4\). Kh. S. Allambergenov proved in his Thesis (Omsk, 1985), that \(c(M_n)=n\) for every \(n\geq 2\).

The authors prove that \(c(M_{23})=2\) completing the set of results of this kind for the class of groups \(\{M_{nt}\}\). It follows that the same assertions are valid for the class \(N_{nt}\), where \(N_{nt}\) denotes the free nilpotent group of rank \(n\) and class \(t\).

The authors also prove that every element of the derived subgroup \(G'\) of a free abelian-by-nilpotent group \(G\) of rank \(n\) can be expressed as a product of \(n\) commutators. Thus by Allambergenov-Roman’kov’s result cited above \(c(G)=n\).

Reviewer: V.A.Roman’kov (Omsk)

### MSC:

20F12 | Commutator calculus |

20F19 | Generalizations of solvable and nilpotent groups |

20E22 | Extensions, wreath products, and other compositions of groups |

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

### Keywords:

free Abelian-by-nilpotent groups; derived subgroups; commutator length; products of commutators; free metabelian groups; free nilpotent groups### Citations:

Zbl 0578.20025
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\textit{M. Akhavan-Malayeri} and \textit{A. Rhemtulla}, Glasg. Math. J. 40, No. 1, 117--121 (1998; Zbl 0911.20028)

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### References:

[1] | Bavard, lndag. Math. N.S. 3 pp 129– (1992) · Zbl 0769.20015 |

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

[3] | Allambergenov, Dokl. Akad. Nauk. UzSSR pp 14– (1984) |

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