A Note on Groups With Many Normal-by-Finite Subgroups

A subgroup X of a group G is called normal-by-finite if it is finite over its core X G XG upper X Subscript upper G . We investigate the behaviour of uncountable groups G of regular cardinality ℵ normal first transfinite cardinal in which all subgroups of cardinality ℵ normal first transfinite cardinal are normal-by-finite. Moreover, generalised soluble groups whose subgroups of infinite rank are either normal-by-finite or contranormal are characterised, and the analogous problem is considered for uncountable groups whose subgroups of large cardinality satisfy the same condition.