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 $X_G$. We investigate the behaviour of uncountable groups $G$ of regular cardinality $\aleph$ in which all subgroups of cardinality $\aleph$ are normal-by-finite. Moreover, generalized soluble groups whose subgroups of infinite rank are either normal-by-finite or contranormal are characterized, and the analogous problem is considered for uncountable groups whose subgroups of large cardinality satisfy the same condition.