doi:10.1007/s00032-012-0172-0engKurdachenko, L. A.Muñoz-Escolano, J. M.Otal, J.An extension of a theorem by B.H. Neumann on groups with boundedly finite conjugacy classesART-2012-79013The 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 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.2012http://zaguan.unizar.es/record/16577610.1007/s00032-012-0172-0http://zaguan.unizar.es/record/165776oai:zaguan.unizar.es:165776Milan Journal of Mathematics 80, 1 (2012), 227-241All rights reservedhttp://www.europeana.eu/rights/rr-f/info:eu-repo/semantics/closedAccess