On Solvable Groups of Arbitrary Derived Length and Small Commutator Length

Let G be a group and G′ its commutator subgroup. Denote by c G the minimal number such that every element ofG′ can be expressed as a product of at most c G commutators. A group G is called a c-group if c G is finite. For any positive integer n, denote by cn the class of groups with commutator length, c G n. Let Fn,t 〈x1, . . . , xn〉 andMn,t 〈x1, . . . , xn〉 be, respectively, the free nilpotent group of rank n and nilpotency class t and the free metabelian nilpotent group of rank n and nilpotency class t. Stroud, in his Ph.D. thesis 1 in 1966, proved that for all t, every element of the commutator subgroup F ′ n,t can be expressed as a product of n commutators. In 1985, Allambergenov and Roman’kov 2 proved that c Mn,t is precisely n, provided that n ≥ 2, t ≥ 4, or n ≥ 3, t ≥ 3. In 3 , Bavard and Meigniez considered the same problem for the n-generator free metabelian group Mn. They showed that the minimum number c Mn of commutators required to express an arbitrary element of the derived subgroup M′ n satisfies [n 2 ] ≤ c Mn ≤ n, 1.1