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 √ n log n. We give a quasipolynomial upper bound, namely, 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 a result of Babai and Seress (1992), our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n. © 2014 Department of Mathematics, Princeton University.