Expressing Elements of the Commutator Subgroup of a Group
as a Product of Least Number of Commutators

The first part of the talk is a survey of present status of the general question: For which groups G generated by d generators is there a bound b = b(d), such that every element in the commutator subgroup [G,G] is expressible as a product of b commutators? This question remains open for finite groups G. It is also open in the case when G is solvable. The solution when G is finite and soluble was obtained in 1999 by D. Segal.

In the second part we turn our attention to the free group F=F(x, y) and give an algorithm and an elementary proof to show that ([x,y])^n can be written as a product of [n/2]+1 commutators and this is the best possible. Here [n/2] is the greatest integer not exceeding n/2. Other related results will also be discussed.