# Sh:971

• Khelif, A., & Shelah, S. (2010). Équivalence élémentaire de puissances cartésiennes d’un même groupe. C. R. Math. Acad. Sci. Paris, 348(23-24), 1241–1244.
• Abstract:
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.
• published version (4p)
Bib entry
@article{Sh:971,
author = {Khelif, Anatole and Shelah, Saharon},
title = {{\'Equivalence \'el\'ementaire de puissances cart\'esiennes d'un m\^eme groupe}},
journal = {C. R. Math. Acad. Sci. Paris},
fjournal = {Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris},
volume = {348},
number = {23-24},
year = {2010},
pages = {1241--1244},
issn = {1631-073X},
mrnumber = {2745331},
mrclass = {20A15 (03B15 03C75 20K99)},
doi = {10.1016/j.crma.2010.10.034}
}