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 \(d\).


20F12 Commutator calculus
20F16 Solvable groups, supersolvable groups
20F05 Generators, relations, and presentations of groups
20F19 Generalizations of solvable and nilpotent groups


Zbl 1030.20017
Full Text: DOI


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