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.