6.39 (1978)

A class of groups $\mathfrak{K}$ is said to be radical if it is closed under taking homomorphic images and normal subgroups and if every group generated by its normal $\mathfrak{K}$-subgroups also belongs to $\mathfrak{K}$. The question proposed relates to the topic “Radical classes and formulae of Narrow (first order) Predicate Calculus (NPC)”. One can show that the only non-trivial radical class definable by universal formulae of NPC is the class of all groups. Recently (in a letter to me), G. M. Bergman constructed a family of locally finite radical classes definable by formulae of NPC. Do there exist similar classes which are not locally finite and different from the class of all groups? In particular, does there exist a radical class of groups which is closed under taking Cartesian products, contains an infinite cyclic group, and is different from the class of all groups?