### 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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