Commutators in metabelian groups. (Commutateurs dans les groupes métabéliens.) (French) Zbl 0769.20015

Let \(G\) be a group and let \(G'\) be the commutator subgroup of \(G\). Then \(c(G)\) is defined as the least positive integer \(k\) such that every element of \(G'\) is the product of \(k\) commutators, if such an integer exists, otherwise \(c(G) = \infty\). The authors prove that if \(G_ n\) is the free metabelian group on \(n\) generators \((n \geq 2)\), then \(c(G_ 2) = 2\) and \(\left[{n\over 2}\right] \leq c(G_ n) \leq n\), where \(\left[{n\over 2}\right]\) is the integral part of \({n\over 2}\). Moreover, if \(G\) is the free metabelian group on a countable set of generators, then \(c(G) = \infty\). As a consequence \(c(G)\) is finite for every polycyclic group \(G\).


20F12 Commutator calculus
20F05 Generators, relations, and presentations of groups
20F16 Solvable groups, supersolvable groups
20E05 Free nonabelian groups
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