A NOTE ON GROUPS WITH MANY NORMAL-BY-FINITE SUBGROUPS

Abstract

A subgroup X of a group G is called normal-by-finite if it is finite over its core

. We investigate the behaviour of uncountable groups G of regular cardinality

ℵ $\aleph $ normal first transfinite cardinal

in which all subgroups of cardinality

ℵ $\aleph $ 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.