Uncountable groups whose large subnormal subgroups are close to normal

Let G be a group. A subgroup X of G is said to be nearly normal if it has finite index in its normal closure XG, and X is called normal-by-finite if it is finite over its core XG. In this paper, we investigate the behavior of uncountable periodic soluble groups G of regular cardinality ℵ in which the section HG/HG is finite (of bounded order) for all subnormal subgroups H of G having cardinality ℵ.