Comparing quasi-finitely axiomatizable and prime groups

An infinite f.g. group G is quasi-finitely axiomatizable (QFA) if there is a first-order sentence j such that G ! j, and if H is a f.g. group such that H ! j, then GGH. The first result is that all Baumslag–Solitar groups of the form ha; d j d"1ad 1⁄4 ai are QFA. A f.g. group G is a prime model if and only if there is a tuple g1; . . . ; gn 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.