On the diameter of finite groups

The diameter of a group G with respect to a set S of generators is the maximum over g∈G of the length of the shortest word in S∪S^-1 representing g. This concept arises in the contexts of efficient communication networks and Rubik´s-cube-type puzzles. `Best´ generators are pertinent to networks, whereas `worst´ and `average´ generators seem more adequate models for puzzles. A substantial body of recent work on these subjects by the authors is surveyed. Regarding the `best´ case, it is shown that, although the structure of the group is essentially irrelevant if |S| is allowed to exceed (log|G |)^1+c(c>0), it plays a strong role when |S|=O(1)