On the diameter of permutation groups
Abstract
Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G,A))$ of the Cayley graph $\Gamma(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{1}$. We are concerned with bounding diam(G):= $\max_A$ diam$(\Gamma(G,A))$. It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$, but the best previously known upper bound was exponential in $\sqrt{n \log n}$. We give a quasipolynomial upper bound, namely, \[\text{diam}(G) = \exp(O((\log n)^4 \log\log n)) = \exp((\log \log G)^{O(1)})\] for G = Sym(n) or G = \Alt(n), where the implied constants are absolute. This addresses a key open case of Babai's conjecture on diameters of simple groups. By standard results, our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree $n$.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 arXiv:
 arXiv:1109.3550
 Bibcode:
 2011arXiv1109.3550H
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Combinatorics;
 Mathematics  Number Theory;
 Mathematics  Probability;
 20B05;
 05C25;
 20B30;
 20F69;
 20D60
 EPrint:
 42 pages. Minimal additions. Last version, to appear in Ann. of Math