ON COMMUTATOR LENGTH OF CERTAIN CLASSES OF SOLVABLE GROUPS

Abstract

Let G be a group and G′ be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. We find suitable bounds for c(G) when G is a free nilpotent by abelian group. Then we prove that c(G) is finite if G is a n-generator solvable group. And G has a nilpotent by abelian normal subgroup K of finite index. Moreover we have c(G) ≤ s(s + 1)/2 + 72n2 + 47n, where s is the number of generators of K. We also prove that in a solvable group of finite Pruffer rank s every element of its commutator subgroup is equal to a product of at most s(s + 1)/2 + 72s2 + 47s. And finally as a corollary of the above results we show that if A is a normal subgroup of a solvable group G such that G/A is a d-generator finite group. And A has finite Pruffer rank s. Then c(G) ≤ s(s + 1)/2 + 72(s2 + n2) + 47(s + n). The bounds we find are independent of the solvability length of the groups.