Bias of group generators in finite and profinite groups: Known results and open problems

We analyze some properties of the distribution QG,k of the first component in a k-tuple chosen uniformly in the set of all the k-tuples generating a finite group G (the limiting distribution of the product replacement algorithm). In particular, we concentrate our attention on the study of the variation distance βk(G) between QG,k and the uniform distribution. We review some known results, analyze several examples and propose some intriguing open questions.