Powers in Alternating Simple Groups

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 Gd = 1
or G = {ad1 ···adN : ai ∈ G}. In a more general context the problem of finding words ω such that the word map (g1, . . . , gd) −→ ω(g1, . . . , gd) 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 x2y2 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.