An Extension of a Theorem by B.H. Neumann on Groups with Boundedly Finite Conjugacy Classes

The source of this paper is a classical theorem by B. H. Neumann on groups whose conjugacy classes are boundedly finite. In a natural way this leads to the study of groups with restrictions on the normal closures of their cyclic subgroups. More concretely, in this paper we study groups G such that the normal closure of every cyclic subgroup ${{\langle{g}\rangle}}$ has a divisible Chernikov G-invariant subgroup D of minimax rank r such that gD has at most b conjugates in the factorgroup G/D. We prove that such groups are Chernikov-by-abelian and bound their invariants in terms of r and b only.