Pronormality in uncountable groups

A subgroup X of a group G is called pronormal if X and Xg are conjugate in < X,X-g > for each element g of G. Pronormal subgroups play a relevant role in many problems of group theory; for instance, they arise naturally in the study of groups in which normality is a transitive relation, because a subgroup is normal if and only if it is subnormal and pronormal. Groups with only pronormal subgroups have been described by Kuzennyi and Subbotin (at least in the locally graded case) and the aim of this paper is to investigate the structure of uncountable locally graded groups of cardinality aleph in which all subgroups of cardinality aleph are pronormal.