Four Variables and Finite Groups

Kerkko Luosto

We consider the strength of the infinitary logic L_{∞ω}^{4} with four variables on finite groups. A simple game-theoretic argument shows that L_{∞ω}^{4} cannot distinguish elementary abelian groups of different dimensions. In contrast, every completely reducible centreless group can be characterized by a single sentence of L_{∞ω}^{4}, and, consequently, by a single sentence of the first order logic L_{ωω}^{4} with four variables, by the result of Dawar, Lindell and Weinstein.

Address: Department of Mathematics P.DO. Box 4 (Yliopistonkatu 5) FIN–00014 University of Helsinki Finland

E-mail: Kerkko.Luosto@Helsinki.Fi