Bias of group generators in the solvable case

Babai and Pak demonstrated a weakness in the product replacement algorithm, a widely used heuristic algorithm intended to rapidly generate nearly uniformly distributed random elements in a finite group G. It was an open question whether the same weakness can be exhibited if one considers only finite solvable groups. We give an affirmative solution to this problem. We consider the distribution of the first component in a k-tuple chosen uniformly in the set of all the k-tuples generating G and construct an infinite sequence of finite solvable groups G for which this distribution turns out to be very far from unifor