Comparing quasi-finitely axiomatizable and prime groups

Abstract

An infinite f.g. group G is quasi-finitely axiomatizable (QFA) if there is a first-order sentence ϕ such that G ⊨ ϕ, and if H is a f.g. group such that H ⊨ ϕ, then G ≅ H . The first result is that all Baumslag–Solitar groups of the form 〈 a, d | d -1 ad = a m 〉 are QFA. A f.g. group G is a prime model if and only if there is a tuple g 1 , … , g n generating G whose orbit (under the automorphisms of G ) is definable by a first-order formula. The second result is that there are continuum many non-isomorphic f.g. groups that are prime models. In particular, not all are QFA.