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

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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.