### 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 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.

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.

Original language | English |
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Pages (from-to) | 251-262 |

Journal | Extracta Mathematicae |

Volume | 30 |

Issue number | 2 |

Publication status | Published - 2015 |

### Keywords

- Alternating groups
- simple groups
- power subgroups
- word maps

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## Cite this

Martinez Carracedo, J. (2015). Powers in Alternating Simple Groups.

*Extracta Mathematicae*,*30*(2), 251-262.