Uncountable groups with restrictions on subgroups of large cardinality

Abstract

The aim of this paper is to investigate the behaviour of uncountable groups of regular cardinality ℵ in which all proper subgroups of cardinality ℵ belong to a given group class X. It is proved that if every proper subgroup of G of cardinality ℵ has finite conjugacy classes, then also the conjugacy classes of G are finite, provided that G has no simple homomorphic images of cardinality ℵ. Moreover, it turns out that if G is a locally graded group of cardinality ℵ in which every proper subgroup of cardinality ℵ contains a nilpotent subgroup of finite index, then G is nilpotent-by-finite, again under the assumption that G has no simple homomorphic images of cardinality ℵ. A similar result holds also for uncountable locally graded groups whose large proper subgroups are abelian-by-finite.