Uncountable groups in which all large subgroups are permutably complemented

A subgroup H of a group G is permutably complemented if it admits a permutable complement, that is, a subgroup K of G such that G= HK and H \cap K =\{1\} . A group G in which every subgroup is permutably complemented is said to be a C -group. The main purpose of this paper is to investigate the behaviour of uncountable groups of regular cardinality \aleph in which all subgroups of cardinality \aleph are permutably complemented, with the particular aim of identifying sufficient conditions for such groups to be C -groups. Finally, in such a context, uncountable soluble groups of cardinality \aleph are considered by restricting attention to only subnormal subgroups of cardinality \aleph whose subnormal defect is at most 2 .