Abstract
The diameter of a group G with respect to a set S of generators is the maximum over g ε G of the length of the shortest word in S 24 S-1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's-cube-type puzzles. 'Best' generators (giving minimum diameter while keeping the number of generators limited) 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). In particular, every non-Abelian finite simple group has a set of ≤7 generators giving logarithmic diameter. This cannot happen for groups with an Abelian subgroup of bounded index. Regarding the worst case, the authors consider permutation groups of degree n and obtain a tight exp((n ln n)1/2(1 + o(1))) upper bound. In the average case, the upper bound improves to exp((ln n)2(1 + O(1))). As a first step toward extending this result to simple groups other than An, it is established that almost every pair of elements of a classical simple group G generates G.
Original language | English |
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Pages (from-to) | 857-865 |
Number of pages | 9 |
Journal | Annual Symposium on Foundations of Computer Science - Proceedings |
Volume | 2 |
State | Published - 1990 |
Externally published | Yes |
Event | Proceedings of the 31st Annual Symposium on Foundations of Computer Science - St. Louis, MO, USA Duration: 22 Oct 1990 → 24 Oct 1990 |