# 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. DOI: 10.1016/j.crma.2010.10.034 MR: 2745331 -
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} }