On solvable groups of arbitrary derived length and small commutator length. (English) Zbl 1234.20045

In a group \(G\), let \(c(G)\) be the minimal number such that every element of \(G'\) can be expressed as a product of at most \(c(G)\) commutators. \(G\) is called \(c\)-group if \(c(G)\) is finite. In particular, for any \(n\geq 1\), \(G\) is a \(c_n\)-group if \(c(G)=n\).
Theorem 2.2 shows that for any \(m\geq 1\) there are solvable groups of derived length \(m\) in which every element of \(G'\) is a commutator. Theorem 2.3 shows that the commutator length of the wreath product of a \(c_1\)-group by an infinite cyclic group is at most 2.


20F16 Solvable groups, supersolvable groups
20F12 Commutator calculus
20F18 Nilpotent groups
20F05 Generators, relations, and presentations of groups
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