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

We analyze some properties of the distribution Q_{G,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 \beta_k(G) between Q_{G,k} and the uniform distribution. We review some known results, analyze several examples and propose some intriguing open questions.

Keywords:

Product replacement algorithm , profinite groups , group generators