Équivalence élémentaire de puissances cartésiennes d'un même groupe

We prove that if I and J are infinite sets and G a commutative torsion group, the groups G^I and G^J are elementarily equivalent for the logic L_∞ω. 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_ω1ω logic. Another argument leads to a countable commutative group having the same property.