Powers in alternating simple groups

Abstract

C. Martínez and E. Zelmanov proved in [12] that for every natural number d and every finite simple group G, there exists a function N = N (d) such that either G d= 1 or G = {a_1^d …a_N^d : a_i ∈ G. In a more general context the problem of finding words ω such that the word map (ɡ1, …, ɡd) → ω (ɡ1, …,ɡd) is surjective for any finite non abelian simple group is a major challenge in Group Theory. In [8] authors give the first example of a word map which is surjective on all finite non-abelian simple groups, the commutator [x; y] (Ore Conjecture). In [11] the conjecture that this is also the case for the word x² y² is formulated. This conjecture was solved in [9] and, independently, in [6], using deep results of algebraic simple groups and representation theory. An elementary proof of this result for alternating simple groups is presented here.