Omega-categorical pseudofinite groups

Abstract

We explore the interplay between omega-categoricity and pseudofiniteness for groups, conjecturing that omega-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of omega-categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the (omega-categorical) model companion of the theory of nilpotent class 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an omega-categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on omega-categorical groups over 50 years.