First Order Classes of Groups Having no Groups With a Given Property

Nataša Božović

A result of Miller [8], that there exists a finitely axiomatizable theory having no nontrivial models with isolvable word problem, is generalized. It is proved here that for every strong hereditary property $P$ of $fp$ group there exist a finitely axiomatizable first-order theory $\Cal I(P)$ having no nontrivial models that enjoy $P$.