On the diameter of finite groups

L. Babai*, G. Hetyei, W. M. Kantor, A. Lubotzky, A. Seress

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

45 Scopus citations

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 languageEnglish
Pages (from-to)857-865
Number of pages9
JournalAnnual Symposium on Foundations of Computer Science - Proceedings
Volume2
StatePublished - 1990
Externally publishedYes
EventProceedings of the 31st Annual Symposium on Foundations of Computer Science - St. Louis, MO, USA
Duration: 22 Oct 199024 Oct 1990

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