Equivalence \'el\'ementaire de puissances cart\'esiennes d'un meme groupe

by Khelif and Shelah. [KhSh:971]
Comptes Rendus de l\'Academie des Sciences, 2010
We prove that if I and J are infinite sets and G an abelian torsion group the groups G^I and G^J are elementarily equivalent for the logic L_{infty omega} . The proof is based on a new and simple property with a Cantor-Bernstein flavour. A criterion applying to non commutative groups allows us to exhibit various groups (free or soluble or nilpotent or ...) G such that for I infinite countable and J uncountable the groups G^I and G^J are not even elementarily equivalent for the L_{omega_I omega} logic. Another argument leads to a countable commutative group having the same property.

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