Groups With All Subgroups Normal-By-Finite

A group G has all of its subgroups normal-by-finite if H/core(G)(H) is finite for all subgroups H of G. These groups can be quite complicated in general, as is seen from the so-called Tarski groups. However, the locally finite groups of this type are shown to be abelian-by-finite; and they are then boundedly core-finite, that is to say, there is a bound depending on G only for the indices \H : core(G)(H)\.