On the diameter of permutation groups

Given a finite group G and a set A of generators, the diameter diam(Γ(G,A)) of the Cayley graph Γ(G,A) is the smallest ℓ such that every element of G can be expressed as a word of length at most ℓ in A∪A−1. We are concerned with bounding diam(G):=maxAdiam(Γ(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 nlogn−−−−−√. We give a quasipolynomial upper bound, namely, diam(G)=exp(O((logn)4loglogn))=exp((loglog