Abstract

We construct explicit generating sets Sn and ˜ Sn of the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), Sn) and C(Sym(n), ˜ Sn) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling and other areas.