Abstract
In this paper we investigate the following problem in Group Theory: which properties $cal P$ transfer (or do not transfer) from all cyclic subgroups, or all abelian subgroups to all arbitrary subgroups? We solve this problem completely when $cal P$ is the property of having finite index in its normal closure, proving that $cal P$ carries from abelian, but not from cyclic to arbitrary subgroups. We use primarily some results of B.H. Neumann.